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09 integration.ipynb
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| "source": "# 09 積分\n\n黒木玄\n\n2018-06-21～2019-04-03\n\n* Copyright 2018 Gen Kuroki\n* License: MIT https://opensource.org/licenses/MIT\n* Repository: https://github.com/genkuroki/Calculus\n\nこのファイルは次の場所できれいに閲覧できる:\n\n* http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/09+integration.ipynb\n\n* https://genkuroki.github.io/documents/Calculus/09+integration.pdf\n\nこのファイルは <a href=\"https://juliabox.com\">Julia Box</a> で利用できる.\n\n自分のパソコンに<a href=\"https://julialang.org/\">Julia言語</a>をインストールしたい場合には\n\n* [WindowsへのJulia言語のインストール](http://nbviewer.jupyter.org/gist/genkuroki/81de23edcae631a995e19a2ecf946a4f)\n\n* [Julia v1.1.0 の Windows 8.1 へのインストール](https://nbviewer.jupyter.org/github/genkuroki/msfd28/blob/master/install.ipynb)\n\nを参照せよ. 前者は古く, 後者の方が新しい.\n\n論理的に完璧な説明をするつもりはない. 細部のいい加減な部分は自分で訂正・修正せよ.\n\n$\n\\newcommand\\eps{\\varepsilon}\n\\newcommand\\ds{\\displaystyle}\n\\newcommand\\Z{{\\mathbb Z}}\n\\newcommand\\R{{\\mathbb R}}\n\\newcommand\\C{{\\mathbb C}}\n\\newcommand\\QED{\\text{□}}\n\\newcommand\\root{\\sqrt}\n\\newcommand\\bra{\\langle}\n\\newcommand\\ket{\\rangle}\n\\newcommand\\d{\\partial}\n\\newcommand\\sinc{\\operatorname{sinc}}\n\\newcommand\\Si{\\operatorname{Si}}\n\\newcommand\\Ci{\\operatorname{Ci}}\n\\newcommand\\si{\\operatorname{si}}\n\\newcommand\\Cin{\\operatorname{Cin}}\n$" | |
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| "source": "<h1>目次<span class=\"tocSkip\"></span></h1>\n<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#積分の基本性質\" data-toc-modified-id=\"積分の基本性質-1\"><span class=\"toc-item-num\">1 </span>積分の基本性質</a></span><ul class=\"toc-item\"><li><span><a href=\"#基本性質(A)線形性,-(B)積分区間の連結,-(C)定数の積分,-(D)単調性\" data-toc-modified-id=\"基本性質(A)線形性,-(B)積分区間の連結,-(C)定数の積分,-(D)単調性-1.1\"><span class=\"toc-item-num\">1.1 </span>基本性質(A)線形性, (B)積分区間の連結, (C)定数の積分, (D)単調性</a></span></li><li><span><a href=\"#性質(E)-積分の絶対値の上からの評価\" data-toc-modified-id=\"性質(E)-積分の絶対値の上からの評価-1.2\"><span class=\"toc-item-num\">1.2 </span>性質(E) 積分の絶対値の上からの評価</a></span></li><li><span><a href=\"#性質(F)-微分積分学の基本定理1\" data-toc-modified-id=\"性質(F)-微分積分学の基本定理1-1.3\"><span class=\"toc-item-num\">1.3 </span>性質(F) 微分積分学の基本定理1</a></span></li><li><span><a href=\"#性質(G)-微分積分学の基本定理2\" data-toc-modified-id=\"性質(G)-微分積分学の基本定理2-1.4\"><span class=\"toc-item-num\">1.4 </span>性質(G) 微分積分学の基本定理2</a></span></li></ul></li><li><span><a href=\"#部分積分と置換積分\" data-toc-modified-id=\"部分積分と置換積分-2\"><span class=\"toc-item-num\">2 </span>部分積分と置換積分</a></span><ul class=\"toc-item\"><li><span><a href=\"#置換積分\" data-toc-modified-id=\"置換積分-2.1\"><span class=\"toc-item-num\">2.1 </span>置換積分</a></span></li><li><span><a href=\"#部分積分\" data-toc-modified-id=\"部分積分-2.2\"><span class=\"toc-item-num\">2.2 </span>部分積分</a></span></li></ul></li><li><span><a href=\"#積分の近似\" data-toc-modified-id=\"積分の近似-3\"><span class=\"toc-item-num\">3 </span>積分の近似</a></span><ul class=\"toc-item\"><li><span><a href=\"#Riemann和\" data-toc-modified-id=\"Riemann和-3.1\"><span class=\"toc-item-num\">3.1 </span>Riemann和</a></span></li><li><span><a href=\"#台形公式\" data-toc-modified-id=\"台形公式-3.2\"><span class=\"toc-item-num\">3.2 </span>台形公式</a></span></li><li><span><a href=\"#Simpsonの公式\" data-toc-modified-id=\"Simpsonの公式-3.3\"><span class=\"toc-item-num\">3.3 </span>Simpsonの公式</a></span></li></ul></li><li><span><a href=\"#広義積分\" data-toc-modified-id=\"広義積分-4\"><span class=\"toc-item-num\">4 </span>広義積分</a></span><ul class=\"toc-item\"><li><span><a href=\"#Fourier変換\" data-toc-modified-id=\"Fourier変換-4.1\"><span class=\"toc-item-num\">4.1 </span>Fourier変換</a></span></li><li><span><a href=\"#絶対収束する広義積分の例\" data-toc-modified-id=\"絶対収束する広義積分の例-4.2\"><span class=\"toc-item-num\">4.2 </span>絶対収束する広義積分の例</a></span><ul class=\"toc-item\"><li><span><a href=\"#arcsin\" data-toc-modified-id=\"arcsin-4.2.1\"><span class=\"toc-item-num\">4.2.1 </span>arcsin</a></span></li><li><span><a href=\"#arctan\" data-toc-modified-id=\"arctan-4.2.2\"><span class=\"toc-item-num\">4.2.2 </span>arctan</a></span></li><li><span><a href=\"#ガンマ函数\" data-toc-modified-id=\"ガンマ函数-4.2.3\"><span class=\"toc-item-num\">4.2.3 </span>ガンマ函数</a></span></li><li><span><a href=\"#ベータ函数\" data-toc-modified-id=\"ベータ函数-4.2.4\"><span class=\"toc-item-num\">4.2.4 </span>ベータ函数</a></span></li><li><span><a href=\"#Gauss分布のFourier変換\" data-toc-modified-id=\"Gauss分布のFourier変換-4.2.5\"><span class=\"toc-item-num\">4.2.5 </span>Gauss分布のFourier変換</a></span></li><li><span><a href=\"#三角函数のLaplace変換\" data-toc-modified-id=\"三角函数のLaplace変換-4.2.6\"><span class=\"toc-item-num\">4.2.6 </span>三角函数のLaplace変換</a></span></li></ul></li><li><span><a href=\"#条件収束する広義積分の例\" data-toc-modified-id=\"条件収束する広義積分の例-4.3\"><span class=\"toc-item-num\">4.3 </span>条件収束する広義積分の例</a></span><ul class=\"toc-item\"><li><span><a href=\"#Dirichlet積分\" data-toc-modified-id=\"Dirichlet積分-4.3.1\"><span class=\"toc-item-num\">4.3.1 </span>Dirichlet積分</a></span></li><li><span><a href=\"#Dirichlet積分の一般化\" data-toc-modified-id=\"Dirichlet積分の一般化-4.3.2\"><span class=\"toc-item-num\">4.3.2 </span>Dirichlet積分の一般化</a></span></li><li><span><a href=\"#Fresnel積分の特殊値\" data-toc-modified-id=\"Fresnel積分の特殊値-4.3.3\"><span class=\"toc-item-num\">4.3.3 </span>Fresnel積分の特殊値</a></span></li><li><span><a href=\"#Dirichelt積分の収束の様子\" data-toc-modified-id=\"Dirichelt積分の収束の様子-4.3.4\"><span class=\"toc-item-num\">4.3.4 </span>Dirichelt積分の収束の様子</a></span></li><li><span><a href=\"#Dirichlet積分が絶対収束しないことの証明\" data-toc-modified-id=\"Dirichlet積分が絶対収束しないことの証明-4.3.5\"><span class=\"toc-item-num\">4.3.5 </span>Dirichlet積分が絶対収束しないことの証明</a></span></li><li><span><a href=\"#Dirichlet積分が絶対収束しない様子\" data-toc-modified-id=\"Dirichlet積分が絶対収束しない様子-4.3.6\"><span class=\"toc-item-num\">4.3.6 </span>Dirichlet積分が絶対収束しない様子</a></span></li><li><span><a href=\"#Fresnel積分\" data-toc-modified-id=\"Fresnel積分-4.3.7\"><span class=\"toc-item-num\">4.3.7 </span>Fresnel積分</a></span></li><li><span><a href=\"#Fresnel積分の収束の様子\" data-toc-modified-id=\"Fresnel積分の収束の様子-4.3.8\"><span class=\"toc-item-num\">4.3.8 </span>Fresnel積分の収束の様子</a></span></li><li><span><a href=\"#数式処理系による定積分の計算\" data-toc-modified-id=\"数式処理系による定積分の計算-4.3.9\"><span class=\"toc-item-num\">4.3.9 </span>数式処理系による定積分の計算</a></span></li><li><span><a href=\"#Dirichlet積分の分母を-$x^a$-とする一般化\" data-toc-modified-id=\"Dirichlet積分の分母を-$x^a$-とする一般化-4.3.10\"><span class=\"toc-item-num\">4.3.10 </span>Dirichlet積分の分母を $x^a$ とする一般化</a></span></li><li><span><a href=\"#部分積分による広義積分の収束の証明法\" data-toc-modified-id=\"部分積分による広義積分の収束の証明法-4.3.11\"><span class=\"toc-item-num\">4.3.11 </span>部分積分による広義積分の収束の証明法</a></span></li></ul></li></ul></li><li><span><a href=\"#積分と極限の交換\" data-toc-modified-id=\"積分と極限の交換-5\"><span class=\"toc-item-num\">5 </span>積分と極限の交換</a></span><ul class=\"toc-item\"><li><span><a href=\"#積分と極限が交換できない場合\" data-toc-modified-id=\"積分と極限が交換できない場合-5.1\"><span class=\"toc-item-num\">5.1 </span>積分と極限が交換できない場合</a></span><ul class=\"toc-item\"><li><span><a href=\"#Lebesgueの収束定理の結論が成立しない場合1\" data-toc-modified-id=\"Lebesgueの収束定理の結論が成立しない場合1-5.1.1\"><span class=\"toc-item-num\">5.1.1 </span>Lebesgueの収束定理の結論が成立しない場合1</a></span></li></ul></li><li><span><a href=\"#Dirichlet積分とその一般化\" data-toc-modified-id=\"Dirichlet積分とその一般化-5.2\"><span class=\"toc-item-num\">5.2 </span>Dirichlet積分とその一般化</a></span><ul class=\"toc-item\"><li><span><a href=\"#Dirichlet積分の一般化に関する公式\" data-toc-modified-id=\"Dirichlet積分の一般化に関する公式-5.2.1\"><span class=\"toc-item-num\">5.2.1 </span>Dirichlet積分の一般化に関する公式</a></span></li><li><span><a href=\"#Dirichlet積分の公式の証明のための準備\" data-toc-modified-id=\"Dirichlet積分の公式の証明のための準備-5.2.2\"><span class=\"toc-item-num\">5.2.2 </span>Dirichlet積分の公式の証明のための準備</a></span></li><li><span><a href=\"#Dirichlet積分の公式の証明\" data-toc-modified-id=\"Dirichlet積分の公式の証明-5.2.3\"><span class=\"toc-item-num\">5.2.3 </span>Dirichlet積分の公式の証明</a></span></li></ul></li><li><span><a href=\"#Fresnel積分の特殊値の公式の導出\" data-toc-modified-id=\"Fresnel積分の特殊値の公式の導出-5.3\"><span class=\"toc-item-num\">5.3 </span>Fresnel積分の特殊値の公式の導出</a></span><ul class=\"toc-item\"><li><span><a href=\"#Fresnel積分の特殊値の公式の証明のための準備-(1/(1+t⁴)の積分)\" data-toc-modified-id=\"Fresnel積分の特殊値の公式の証明のための準備-(1/(1+t⁴)の積分)-5.3.1\"><span class=\"toc-item-num\">5.3.1 </span>Fresnel積分の特殊値の公式の証明のための準備 (1/(1+t⁴)の積分)</a></span></li><li><span><a href=\"#1/(1+t⁴)の不定積分の公式の数式処理を用いた確認\" data-toc-modified-id=\"1/(1+t⁴)の不定積分の公式の数式処理を用いた確認-5.3.2\"><span class=\"toc-item-num\">5.3.2 </span>1/(1+t⁴)の不定積分の公式の数式処理を用いた確認</a></span></li><li><span><a href=\"#Fresnel積分の特殊値の公式の証明\" data-toc-modified-id=\"Fresnel積分の特殊値の公式の証明-5.3.3\"><span class=\"toc-item-num\">5.3.3 </span>Fresnel積分の特殊値の公式の証明</a></span></li><li><span><a href=\"#Fresnel積分の特殊値の公式の複素解析を用いた導出\" data-toc-modified-id=\"Fresnel積分の特殊値の公式の複素解析を用いた導出-5.3.4\"><span class=\"toc-item-num\">5.3.4 </span>Fresnel積分の特殊値の公式の複素解析を用いた導出</a></span></li><li><span><a href=\"#sine-integral-and-cosine-integral-functions\" data-toc-modified-id=\"sine-integral-and-cosine-integral-functions-5.3.5\"><span class=\"toc-item-num\">5.3.5 </span>sine integral and cosine integral functions</a></span></li></ul></li></ul></li></ul></div>" | |
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| "source": "using Base.MathConstants\nusing Base64\nusing Printf\nusing Statistics\nconst e = ℯ\nendof(a) = lastindex(a)\nlinspace(start, stop, length) = range(start, stop, length=length)\n\nusing Plots\ngr(); ENV[\"PLOTS_TEST\"] = \"true\"\n#clibrary(:colorcet)\nclibrary(:misc)\n\nfunction pngplot(P...; kwargs...)\n sleep(0.1)\n pngfile = tempname() * \".png\"\n savefig(plot(P...; kwargs...), pngfile)\n showimg(\"image/png\", pngfile)\nend\npngplot(; kwargs...) = pngplot(plot!(; kwargs...))\n\nshowimg(mime, fn) = open(fn) do f\n base64 = base64encode(f)\n display(\"text/html\", \"\"\"<img src=\"data:$mime;base64,$base64\">\"\"\")\nend\n\nusing SymPy\n#sympy.init_printing(order=\"lex\") # default\n#sympy.init_printing(order=\"rev-lex\")\n\nconst latex = sympy.latex\nusing LaTeXStrings\nlatexstring_displaystyle(args...; kwargs...) = \n LaTeXString(raw\"$$\" * prod(latex.(args; kwargs...)) * raw\"$$\")\nlatexdisp(args...; kwargs...) = \n display(latexstring_displaystyle(args...; kwargs...))\nconst ls = latexstring_displaystyle\nconst ld = latexdisp\n\nusing SpecialFunctions\nusing QuadGK", | |
| "execution_count": 1, | |
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| }, | |
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| "cell_type": "markdown", | |
| "source": "## 積分の基本性質" | |
| }, | |
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| "source": "1変数函数 $f(x)$ の積分 $\\int_a^b f(x)\\,dx$ の構成の仕方は無数に存在する.\n\nRiemann積分とLebesgue積分は有名であり, それら以外にも様々な方法が存在する. 例えば, \n\n* S. ラング, 『ラング 現代の解析学』, 共立出版, 1981 (原書: Serge Lang, Real Analysis, 1969)\n\nの第5章「微分法」の第1節では, \n\n$$\nf(t) = v_i \\quad (a_{i-1}<t<a_i,\\ a=a_0\\leqq a_1\\leqq\\cdots\\leqq a_n=b)\n$$\n\nを満たす函数(階段函数と呼ぶことにする)の積分を\n\n$$\n\\int_a^b f(t)\\,dt = \\sum_{i=1}^n v_i(a_i-a_{i-1})\n$$\n\nと定め, これを階段函数の一様収束先に拡張することによって, 積分を導入している. 閉区間上の連続函数は階段函数の一様収束先になっているので, 連続函数の積分もこれによって定義されることになる.\n\n積分の構成の仕方は無数にあるが, 積分は構成の仕方によらない基本性質を持つ. だから, その基本性質のみを使って示された結果はどの積分論においても利用可能になる.\n\nこのノートでは積分の構成の仕方には深く踏み込まずに, 積分の基本性質のみを使って積分論を展開する. \n\nただし, 積分の基本性質の厳密な説明を避けて, 大らかなスタイルで議論を進めることにする." | |
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| "source": "### 基本性質(A)線形性, (B)積分区間の連結, (C)定数の積分, (D)単調性\n\n**基本性質(A)** **線形性:** 定数 $\\alpha,\\beta$ について\n\n$$\n\\int_a^b(\\alpha f(x)+\\beta g(x))\\,dx = \\alpha\\int_a^b f(x)\\,dx + \\beta\\int_a^b g(x)\\,dx.\n$$\n\n**基本性質(B)** **積分区間の連結:**\n\n$$\n\\int_a^b f(x)\\,dx + \\int_b^c\\,f(x)\\,dx = \\int_a^c f(x)\\,dx.\n$$\n\n**注意:** (1) (B)において $a=b=c$ と仮定すると $\\int_a^a f(x)\\,dx=0$ が導かれる. \n\n(2) (B)において $c=a$ と仮定すると $\\int_b^a f(x)\\,dx = - \\int_a^b f(x)\\,dx$ が導かれる.\n\n(3) ゆえに $a\\leqq b$ のときに $\\int_a^b f(x)\\,dx$ が定義されていれば, $a > b$ の場合には $\\int_a^b f(x)\\,dx$ を $\\int_a^b f(x)\\,dx = -\\int_b^a f(x)\\,dx$ で定義することによって, $a,b$ の大小に条件を付ける必要がなくなる.\n\n(4) (B)は $\\ds\\int_a^c f(x)\\,dx - \\int_b^a\\,f(x)\\,dx = \\int_b^c f(x)\\,dx$ と同値である. 基本性質(B)はこの形でもよく使われる.\n\n**基本性質(C)** **定数の積分:** 定数 $\\alpha$ について,\n\n$$\n\\int_a^b \\alpha\\,dx = \\alpha(b-a).\n$$\n\n**注意:** (1) $\\int_a^b 1\\,dx = b-a$ と(A)から(C)が導かれる.\n\n(2) 函数 $f(x)$ が $a<x<b$ で一定の値 $\\alpha$ の値を取るならば(すなわち $f(x)=\\alpha$ ($a<x<b$)) ならば $\\int_a^b f(x)\\,dx = \\alpha(b-a)$ となるという(C)の一般化を仮定すると, (B)と合わせて, 階段函数の積分が確定する. \n\n**基本性質(D)** **単調性:** $a\\leqq b$ と仮定する. 1変数実数値函数 $f(x)$, $g(x)$ について\n\n$$\nf(x)\\leqq g(x)\\quad (a<x<b) \\quad\\implies\\quad \\int_a^b f(x)\\,dx\\leqq\\int_a^b g(x)\\,dx.\n$$" | |
| }, | |
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| "source": "### 性質(E) 積分の絶対値の上からの評価\n\n**(E)** $a\\leqq b$ と仮定する. 1変数実数値函数 $f(x)$ について\n\n$$\n\\left|\\int_a^b f(x)\\,dx\\right| \\leqq \\int_a^b |f(x)|\\,dx.\n\\tag{1}\n$$\n\nさらに $a\\leqq x\\leqq b$ で $|f(x)|\\leqq M$ ならば\n\n$$\n\\left|\\int_a^b f(x)\\,dx\\right| \\leqq M|b-a|.\n\\tag{2}\n$$\n\n**注意:** (2)は $a>b$ であっても成立している. そのために絶対値を付けた. \n\n**証明:** $-|f(x)|\\leqq f(x)\\leqq |f(x)|$ なので積分の単調性(D)より\n\n$$\n-\\int_a^b|f(x)|\\,dx \\leqq \\int_a^b f(x)\\,dx \\leqq \\int_a^b |f(x)|\\,dx.\n$$\n\nこれは(1)が成立することを意味する. $a\\leqq x\\leqq b$ で $|f(x)|\\leqq M$ ならば積分の単調性(D)と(C)より,\n\n$$\n\\int_a^b |f(x)|\\,dx \\leqq \\int_a^b M\\,dx = M(b-a)=M|b-a|.\n$$\n\nこれで(2)も成立することが示された. $\\QED$\n\n**解説:** 多くの場面で「小さいこと」を示したい誤差項は積分で表わされる. (Taylorの定理における積分型剰余項はその典型例である.) 成立(E)は積分の絶対値が「小さいこと」を示すためによく使われることになる. $\\QED$" | |
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| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**注意:** $f(x)$ が複素数値函数であっても性質(E)は成立している. さらに絶対値を任意のノルムで置き換えればベクトル値函数であっても性質(E)は成立している. しかし, そのことを示すためには性質(A)～(D)を仮定するだけでは足りないので, 複素数値函数やベクトル値函数まで拡張された性質(E)も必要ならば, 最初から拡張された性質(E)も仮定しておく方がよいかもしれない.\n\n拡張された性質(E)が主要などの積分論でも成立する理由は, どの積分論においても積分が有限和 $v_1+\\cdots+v_n$ の極限で構成されているからである. ここで $v_i$ は複素数またはベクトルである. 絶対値およびより一般にノルムは三角不等式\n\n$$\n\\|v_1+\\cdots+v_n\\| \\leqq \\|v_1\\|+\\cdots+\\|v_n\\|\n$$\n\nを満たしている. これの極限として複素数値もしくはベクトル値函数 $f(x)$ についても\n\n$$\n\\left\\|\\int_a^b f(x)\\,dx\\right\\|\\leqq \\int_a^b \\|f(x)\\|\\,dx\n$$\n\nが成立することが示される. この不等式を点の運動 $x(t)$ の速度ベクトル $v(t)=x'(t)$ に適用した結果は\n\n$$\n\\|x(b)-x(a)\\|=\\left\\|\\int_a^b v(t)\\,dt\\right\\|\\leqq \\int_a^b \\|v(t)\\|\\,dt\n$$\n\nとなる. 速さ $\\|v(t)\\|=\\|x'(t)\\|$ の時刻 $a$ から $b$ までの積分の値は点の運動の軌跡(曲線)の長さなので, この不等式は $x(a)$ と $x(b)$ の距離は $x(a)$ と $x(b)$ を結ぶ曲線の長さ以下になることを意味する. \n\nこれより, ベクトル値函数に拡張された性質(E)は, 直観的には当然そうなるべき結果を意味していることがわかる. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 性質(F) 微分積分学の基本定理1\n\n**(F)** 微分積分学の基本定理1: 連続函数 $f(x)$ について\n\n$$\n\\frac{d}{dx}\\int_a^x f(\\xi)\\,d\\xi = f(x).\n$$\n\n**証明:** $F(x)=\\int_a^x f(\\xi)\\,d\\xi$ とおく. このとき(B)より\n\n$$\nF(x+h)-F(x) = \\int_a^{x+h}f(\\xi)\\,d\\xi - \\int_a^x f(\\xi)\\,d\\xi = \\int_x^{x+h}f(\\xi)\\,d\\xi.\n$$\n\nさらに $f(x)$ は積分変数 $\\xi$ について定数なので, (C)より\n\n$$\nf(x)h = f(x)((x+h) - x) = \\int_x^{x+h} f(x)\\,d\\xi.\n$$\n\nゆえに\n\n$$\nF(x+h)-F(x)-f(x)h = \n\\int_x^{x+h}f(\\xi)\\,d\\xi - \\int_x^{x+h} f(x)\\,d\\xi =\n\\int_x^{x+h}(f(\\xi) - f(x))\\,d\\xi.\n$$\n\n任意の $\\eps > 0$ を取る. $f(\\xi)$ は $\\xi$ の連続函数なので $|h|$ を十分に小さくすると\n\n$$\n|f(\\xi)-f(x)|\\leqq \\eps\n$$\n\nとなる(図を描いてみよ). ゆえに(E)より\n\n$$\n|F(x+h)-F(x)-f(x)h|=\\left|\\int_x^{x+h}(f(\\xi) - f(x))\\,d\\xi\\right|\n\\leqq \\eps|(x+h)-x| = \\eps|h|.\n$$\n\n両辺を $|h|$ で割ることによって, $\\eps>0$ ごとに $|h|>0$ を十分に小さくすることによって, \n\n$$\n\\left|\\frac{F(x+h)-F(x)-f(x)h}{h}\\right|\\leqq \\eps\n$$\n\nとなることわかる. これは $h\\to 0$ のとき\n\n$$\n\\frac{F(x+h)-F(x)}{h}-f(x) = \\frac{F(x+h)-F(x)-f(x)h}{h} \\to 0\n$$\n\nが成立することを意味する. これで $F'(x) = f(x)$ となることを示せた. $\\QED$\n\n**注意:** 上の証明では $\\eps$-$\\delta$ 論法を使った. $\\eps$-$\\delta$ 論法を未習もしくはまだ理解していない人であっても, $f(x)$ のグラフを適当に描いて, 積分が「面積」を表すことを思い出しながら, 証明の各ステップに現われる積分がグラフのどの部分の「面積」になっているかを確認すれば, 直観的には明らかな議論しか行っていないことがわかるだろう. むしろ $\\eps$-$\\delta$ 論法を使った証明を使いこなせるようになるためには, $\\eps$-$\\delta$ 論法を理解していない段階でのそのような直観的な議論を十分にしておいた方がよい. 慣れてしまえば, $\\eps$-$\\delta$ 論法自体が直観的に明らかな議論に見えて来ることになるだろう. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 性質(G) 微分積分学の基本定理2\n\n**準備:** 微分可能函数 $f(x)$ の導函数 $f'(x)$ が恒等的に $0$ ならば $f(x)$ は定数函数になることを(G)の証明では認めて使うことにする. その結果は平均値の定理\n\n$$\nf(x) - f(a) = f'(\\xi)(x-a) \\quad (\\xi = a+t(x-a),\\ 0<t<1)\n$$\n\nを使えば容易に導かれる($\\xi$ は $a$, $x$ のあいだのある実数である). $f'$ が恒等的に $0$ ならば特に $f'(\\xi)=0$ となるので $f(x)=f(a)$ となり, $f(x)$ の値は $x$ の値によらず $f(a)$ に等しくなる. $\\QED$\n\n**(G)** 微分積分学の基本定理2: $F(x)$ は $C^1$ 級函数(微分可能かつ導函数が連続な函数)であるとすると, \n\n$$\n\\int_a^b F'(x)\\,dx = F(b) - F(a).\n\\tag{1}\n$$\n\n**注意:** (1)の右辺を $[F(x)]_a^b$ と書くことが多い:\n\n$$\n[F(x)]_a^b = F(b) - F(a).\n$$\n\n**証明:** $G(x)=\\int_a^x F'(\\xi)\\,d\\xi - (F(x)-F(a))$ とおく. このとき, (B)より\n\n$$\nG(a) = \\int_a^a F'(\\xi)\\,d\\xi - (F(a)-F(a)) = 0\n$$\n\nとなり, (F)より\n\n$$\nG'(x) = F'(x) - F'(x) = 0.\n$$\n\nゆえに $G(x)$ は定数函数になり, 恒等的に $0$ になる. ゆえに\n\n$$\n0 = G(b) = \\int_a^b F'(x)\\,dx - (F(b)-F(a)).\n$$\n\nこれは(1)が成立することを意味する. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**以上によって, 実1変数実数値函数の微分積分学の基本定理が積分の基本性質(A)～(D)だけから導かれることがわかった.**\n\n**注意:** 実1変数の複素数値函数もしくはベクトル値函数に関する微分積分学の基本定理は積分の基本性質(A)～(D)と拡張された性質(E)のみを使って導き出せる. 性質(A)～(E)は高校数学レベルの積分に関する直観があれば「明らか」だと感じられるシンプルな性質の集まりである. そのような「明らか」でかつシンプルな性質のみを使って微分積分学の基本定理は導出可能なのである. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "## 部分積分と置換積分\n\n微分積分学の基本定理さえ証明できてしまえば, 部分積分と置換積分の導出は微分学の結果をそのまま積分学に応用する話に過ぎなくなる." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 置換積分\n\n$f(x)$, $g(t)$ は $C^1$ 級函数であると仮定し, \n\n$$\nF(x) = \\int_a^x f(\\xi)\\,d\\xi\n$$\n\nとおく. 微分の性質\n\n$$\n\\frac{d}{dt}F(g(t)) = F'(g(t))g'(t) = f(g(t))g'(t)\n$$\n\nに微分積分学の基本定理(G)を適用すると, $a=g(\\alpha)$, $b=g(\\beta)$ のとき, \n\n$$\n\\int_a^b f(x)\\,dx = [F(x)]_a^b = [F(g(t))]_\\alpha^\\beta =\n\\int_\\alpha^\\beta \\frac{d}{dt}F(g(t))\\,dt = \\int_\\alpha^\\beta f(g(t))g'(t)\\,dt.\n$$\n\nつまり, 次が成立している.\n\n**(H)** **置換積分:** $a=g(\\alpha)$, $b=g(\\beta)$ のとき,\n\n$$\n\\int_a^b f(x)\\,dx = \\int_\\alpha^\\beta f(g(t))g'(t)\\,dt.\n$$\n\n左辺の積分を右辺で計算することを**置換積分**と呼ぶ. $x(t)=g(t)$ とおいて,\n\n$$\n\\int_a^b f(x)\\,dx = \\int_\\alpha^\\beta f(x(t))\\frac{dx(t)}{dt}\\,dt\n$$\n\nと書き直すと, 「分子分母の $dt$ が約分で消える公式」のように見えて見易い.\n\n**問題:** 任意の微積分の教科書を参照して, 置換積分を使って計算する積分計算の面白い問題を見付けて, 計算練習せよ. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 部分積分\n\n$f(x)$, $g(x)$ は $C^1$ 級函数であると仮定する.\n\n微分の性質\n\n$$\n(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)\n$$\n\nに微分積分学の基本定理(G)と積分の線形性(A)を適用すると\n\n$$\n[f(x)g(x)]_a^b = \\int_a^b (f(x)g(x))'\\,dx = \\int_a^b f'(x)g(x)\\,dx + \\int_a^b f(x)g'(x)\\,dx.\n$$\n\nこれは以下と同値である.\n\n**(I)** **部分積分:** $f(x)$, $g(x)$ は $C^1$ 級函数であるとき, \n\n$$\n\\begin{aligned}\n&\n\\int_a^b f'(x)g(x)\\,dx = [f(x)g(x)]_a^b - \\int_a^b f(x)g'(x)\\,dx, \n\\\\ &\n\\int_a^b f(x)g'(x)\\,dx = [f(x)g(x)]_a^b - \\int_a^b f'(x)g(x)\\,dx.\n\\end{aligned}\n$$\n\n左辺の積分を右辺で計算することを**部分積分**と呼ぶ.\n\n**問題:** 任意の微積分の教科書を参照して, 部分積分を使って計算する積分計算の面白い問題を見付けて, 計算練習せよ. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "## 積分の近似" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "$a<b$ であると仮定し, $f$ は閉区間 $[a,b]$ 上の連続函数であると仮定する." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### Riemann和\n\n閉区間 $[a,b]$ の分割 $a=x_0\\leqq x_1\\leqq\\cdots\\leqq x_N=b$ と $x_i^*\\in[x_{i-1},x_i]$ と $\\Delta x_i=x_i-x_{i-1}$ に対して, 和\n\n$$\n\\sum_{i=1}^N f(x_i^*)\\Delta x_i = \\sum_{i=1}^N f(x_i^*)(x_i-x_{i-1})\n$$\n\nを**Riemann和**と呼ぶ. $\\Delta x_i$ の最大値を小さくすることによって(分割を細かくすることによって), Riemann和で積分 $\\int_a^b f(x)\\,dx$ を幾らでも近似できることを積分の基本性質のみを使って証明しよう.\n\n**証明:** 任意の $\\eps>0$ を取る. $f(x)$ は閉区間 $[a,b]$ 上の連続函数であると仮定してあったので, $f(x)$ は閉区間 $[a,b]$ 上の一様連続函数になる. $\\Delta x_i$ 達の最大値を十分小さくすれば\n\n$$\n|f(x)-f(x_i^*)| \\leqq \\frac{\\eps}{b-a}\\quad (x_{i-1}\\leqq x\\leqq x_i)\n$$\n\nとなる. したがって, \n\n$$\n\\begin{aligned}\n\\left|\\sum_{i=1}^N f(x_i^*)\\Delta x_i-\\int_a^b f(x)\\,dx\\right| &=\n\\left|\\sum_{i=1}^N \\int_{x_{i-1}}^{x_i}f(x_i^*)\\,dx-\\sum_{i=1}^N\\int_{x_{i-1}}^{x_i} f(x)\\,dx\\right|\n\\\\ &=\n\\left|\\sum_{i=1}^N \\int_{x_{i-1}}^{x_i}(f(x_i^*)-f(x))\\,dx\\right|\n\\\\ &\\leqq\n\\sum_{i=1}^N \\left|\\int_{x_{i-1}}^{x_i}(f(x_i^*)-f(x))\\,dx\\right|\n\\\\ &\\leqq\n\\sum_{i=1}^N \\int_{x_{i-1}}^{x_i}|f(x_i^*)-f(x)|\\,dx\n\\\\ &\\leqq\n\\sum_{i=1}^N \\int_{x_{i-1}}^{x_i}\\frac{\\eps}{b-a}\\,dx =\n\\sum_{i=1}^N \\frac{\\eps}{b-a}\\Delta x_i \n\\\\ &= \n\\frac{\\eps}{b-a}(b-a) = \n\\eps\n\\end{aligned}\n$$\n\n以上によって, 閉区間 $[a,b]$ の分割を細かくすることによって, Riemann和で積分を誤差 $\\eps$ 以下で近似できることを示せた. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**定理:** 閉区間 $[a,b]$ の分割のデータ\n\n$$\na=x_0\\leqq x_1\\leqq\\cdots\\leqq x_N=b, \\quad x_i^*\\in[x_{i-1},x_i], \\quad \\Delta x_i=x_i-x_{i-1}\n$$\n\nにおいて, $\\Delta x_i$ の最大値を $0$ に近付ける極限を, $\\ds\\lim_{\\max\\{\\Delta x_i\\}\\searrow 0}$ と書くと, $[a,b]$ 上の連続函数 $f(x)$ について,\n\n$$\n\\int_a^b f(x)\\,dx = \\lim_{\\max\\{\\Delta x_i\\}\\searrow 0} \\sum_{i=1}^N f(x_i^*)\\Delta x_i\n$$\n\nが成立する. $\\QED$\n\n**注意(Riemann積分):** 積分の基本性質とは無関係に, 閉区間 $[a,b]$ 上の連続函数 $f(x)$ に対して, 極限\n\n$$\n\\lim_{\\max\\{\\Delta x_i\\}\\searrow 0} \\sum_{i=1}^N f(x_i^*)\\Delta x_i\n$$\n\nが存在することを示せる. この極限が存在するような閉区間 $[a,b]$ 上の函数は**Riemann積分可能**な函数と呼ばれ, その極限の値は**Riemann積分**と呼ばれる. 特に閉区間上の連続函数はRiemann積分可能である. そして, Riemann積分で積分を定義すると, Riemann積分可能な函数について, 積分の基本性質をすべて示すことができる. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 台形公式\n\n閉区間 $[a,b]$ の分割 $a=x_0\\leqq x_1\\leqq\\cdots\\leqq x_N=b$ に対する和\n\n$$\n\\sum_{i=1}^N \\frac{f(x_{i-1})+f(x_i)}{2}\\Delta x_i\n$$\n\nで積分 $\\int_a^b f(x)\\,dx$ を近似することもできる. これを**台形公式**(trapezoidal rule)と呼ぶ." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### Simpsonの公式\n\n閉区間 $[a,b]$ の分割 $a=x_0\\leqq x_1\\leqq\\cdots\\leqq x_N=b$ に対する和\n\n$$\n\\sum_{i=1}^N \\frac{f(x_{i-1}) + 4\\,f((x_{i-1}+x_i)/2) + f(x_i)}{6}\\Delta x_i\n$$\n\nで積分 $\\int_a^b f(x)\\,dx$ を近似することもできる. これを**Simpsonの公式**(Simpson's rule)と呼ぶ." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**Simpsonの公式の由来:** $a<b$ であるとし, $c=(a+b)/2$ とおく. このとき, $f(a)=\\alpha$, $f(c)=\\gamma$, $f(b)=\\beta$ を満たす2次以下の多項式 $f(x)$ が一意的に決まり, \n\n$$\nf(x) = \n\\alpha\\frac{(x-b)(x-c)}{(a-b)(a-c)} + \n\\beta \\frac{(x-a)(x-c)}{(b-a)(b-c)} + \n\\gamma\\frac{(x-a)(x-b)}{(c-a)(c-b)} \n$$\n\nと一意に表わされる. このとき,\n\n$$\n\\begin{aligned}\n&\n\\int_a^b \\frac{(x-b)(x-c)}{(a-b)(a-c)}\\,dx = \\frac{1}{6}(b-a), \n\\\\ &\n\\int_a^b \\frac{(x-a)(x-c)}{(b-a)(b-c)}\\,dx = \\frac{1}{6}(b-a), \n\\\\ &\n\\int_a^b \\frac{(x-a)(x-b)}{(c-a)(c-b)}\\,dx = \\frac{4}{6}(b-a)\n\\end{aligned}\n$$\n\nとなることを確認できるので\n\n$$\n\\int_a^b f(x)\\,dx = \\frac{f(a)+4f((a+b)/2)+f(b)}{6}(b-a)\n$$\n\nとなることがわかる.\n\nつまり, $f(x)$ が2次函数のとき閉区間 $[a,b]$ の自明な分割 $a=a_0<a_1=b$ に関するSimpsonの公式は誤差無しで積分の値を計算する公式になっている. $\\QED$\n\nSimpsonの公式の他の解釈については\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20170724EulerMaclaurin.pdf\">Euler-Maclaurinの和公式の一般化</a>\n\nの第3.5と第3.6節を参照せよ." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題:** 上のSimpsonの公式の由来の解説において結果だけを書いた積分計算を自分で実行せよ. $\\QED$\n\n解答略." | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "a, b, c, x = symbols(\"a b c x\", real=true)\n\nI1 = sympy.Integral((x-b)*(x-c)/((a-b)*(a-c)), (x,a,b))\nI2 = sympy.Integral((x-a)*(x-c)/((b-a)*(b-c)), (x,a,b))\nI3 = sympy.Integral((x-a)*(x-b)/((c-a)*(c-b)), (x,a,b))\n\nld(c, \" = \", factor((a+b)/2))\nld(I1, \" = \", I1(c=>(a+b)/2).doit().simplify().factor())\nld(I2, \" = \", I2(c=>(a+b)/2).doit().simplify().factor())\nld(I3, \" = \", I3(c=>(a+b)/2).doit().simplify().factor())", | |
| "execution_count": 2, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$c = \\frac{a + b}{2}$$\"", | |
| "text/latex": "$$c = \\frac{a + b}{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{a}^{b} \\frac{\\left(- b + x\\right) \\left(- c + x\\right)}{\\left(a - b\\right) \\left(a - c\\right)}\\, dx = - \\frac{a - b}{6}$$\"", | |
| "text/latex": "$$\\int_{a}^{b} \\frac{\\left(- b + x\\right) \\left(- c + x\\right)}{\\left(a - b\\right) \\left(a - c\\right)}\\, dx = - \\frac{a - b}{6}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{a}^{b} \\frac{\\left(- a + x\\right) \\left(- c + x\\right)}{\\left(- a + b\\right) \\left(b - c\\right)}\\, dx = - \\frac{a - b}{6}$$\"", | |
| "text/latex": "$$\\int_{a}^{b} \\frac{\\left(- a + x\\right) \\left(- c + x\\right)}{\\left(- a + b\\right) \\left(b - c\\right)}\\, dx = - \\frac{a - b}{6}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{a}^{b} \\frac{\\left(- a + x\\right) \\left(- b + x\\right)}{\\left(- a + c\\right) \\left(- b + c\\right)}\\, dx = - \\frac{2 \\left(a - b\\right)}{3}$$\"", | |
| "text/latex": "$$\\int_{a}^{b} \\frac{\\left(- a + x\\right) \\left(- b + x\\right)}{\\left(- a + c\\right) \\left(- b + c\\right)}\\, dx = - \\frac{2 \\left(a - b\\right)}{3}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
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| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題:** Simpsonの公式の由来の説明に基いて, 以下の公式を導け.\n\n$$\n\\int_a^b (x-a)(x-b)\\,dx = -\\frac{(b-a)^3}{6}, \\quad\n\\int_a^b (x-a)\\left(x-\\tfrac{a+b}{2}\\right)\\,dx = \\frac{(b-a)^3}{12}.\n\\qquad \\QED\n$$\n\n解答略." | |
| }, | |
| { | |
| "metadata": { | |
| "code_folding": [], | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "a, b, x = symbols(\"a b x\", real=true)\nc = factor((a+b)/2)\n\nI1 = sympy.Integral((x-a)*(x-b), (x,a,b))\nI2 = sympy.Integral((x-a)*(x-c), (x,a,b))\n\nld(I1, \" = \", I1.doit().simplify().factor())\nld(I2, \" = \", I2.doit().simplify().factor())", | |
| "execution_count": 3, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{a}^{b} \\left(- a + x\\right) \\left(- b + x\\right)\\, dx = \\frac{\\left(a - b\\right)^{3}}{6}$$\"", | |
| "text/latex": "$$\\int_{a}^{b} \\left(- a + x\\right) \\left(- b + x\\right)\\, dx = \\frac{\\left(a - b\\right)^{3}}{6}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{a}^{b} \\left(- a + x\\right) \\left(x - \\frac{a + b}{2}\\right)\\, dx = - \\frac{\\left(a - b\\right)^{3}}{12}$$\"", | |
| "text/latex": "$$\\int_{a}^{b} \\left(- a + x\\right) \\left(x - \\frac{a + b}{2}\\right)\\, dx = - \\frac{\\left(a - b\\right)^{3}}{12}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題:** 円周率を表わす2種類の積分\n\n$$\n4\\int_0^1\\sqrt{1-x^2}\\,dx = \\pi, \\quad\n4\\int_0^1\\frac{dx}{1+x^2} = \\pi\n$$\n\nの近似計算を, 閉区間 $[0,1]$ の分割 $x_i = i/N$ ($i=0,1,\\ldots,N$) と $x_i^*=(x_{i-1}+x_i)/2$ に関するRiemann和と台形公式とSimpsonの公式を用いて実行してみよ. $N=10$ の場合にどうなるか? $\\QED$" | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "function RiemannSum(f, a, b, N)\n Δx = (b-a)/N\n sum(i->f(a+(i-1/2)*Δx)*Δx, 1:N)\nend\n\nfunction TrapezoidalRule(f, a, b, N)\n Δx = (b-a)/N\n sum(i->(f(a+(i-1)*Δx)+f(a+i*Δx))/2*Δx, 1:N)\nend\n\nfunction SimpsonRule(f, a, b, N)\n Δx = (b-a)/N\n sum(i->(f(a+(i-1)*Δx)+4f(a+(i-1/2)*Δx)+f(a+i*Δx))/6*Δx, 1:N)\nend\n\nfunction SymPyIntegrate(f, a, b)\n x = symbols(\"x\", real=true)\n integrate(f(x), (x, a, b))\nend", | |
| "execution_count": 4, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 4, | |
| "data": { | |
| "text/plain": "SymPyIntegrate (generic function with 1 method)" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "f(x) = 4x + 3\na, b = 0.0, 1.0\nN = 10\n\n@show RiemannSum(f, a, b, N)\n@show TrapezoidalRule(f, a, b, N)\n@show SimpsonRule(f, a, b, N)\n@show s = SymPyIntegrate(f, Sym(0), Sym(1))\nfloat(s)", | |
| "execution_count": 5, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "RiemannSum(f, a, b, N) = 5.0\nTrapezoidalRule(f, a, b, N) = 5.0\nSimpsonRule(f, a, b, N) = 5.0\ns = SymPyIntegrate(f, Sym(0), Sym(1)) = 5\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 5, | |
| "data": { | |
| "text/plain": "5.0" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "f(x) = 2x^2 - x + 3\na, b = 0.0, 1.0\nN = 10\n\n@show RiemannSum(f, a, b, N)\n@show TrapezoidalRule(f, a, b, N)\n@show SimpsonRule(f, a, b, N)\n@show s = SymPyIntegrate(f, Sym(0), Sym(1))\nfloat(s)", | |
| "execution_count": 6, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "RiemannSum(f, a, b, N) = 3.1650000000000005\nTrapezoidalRule(f, a, b, N) = 3.170000000000001\nSimpsonRule(f, a, b, N) = 3.166666666666667\ns = SymPyIntegrate(f, Sym(0), Sym(1)) = 19/6\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 6, | |
| "data": { | |
| "text/plain": "3.1666666666666665" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "f(x) = 4*√(1-x^2)\na, b = 0.0, 1.0\nN = 10\n\n@show RiemannSum(f, a, b, N)\n@show TrapezoidalRule(f, a, b, N)\n@show SimpsonRule(f, a, b, N)\n@show s = SymPyIntegrate(f, Sym(0), Sym(1))\nfloat(s)", | |
| "execution_count": 7, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "RiemannSum(f, a, b, N) = 3.1524114332616446\nTrapezoidalRule(f, a, b, N) = 3.1045183262483182\nSimpsonRule(f, a, b, N) = 3.136447064257202\ns = SymPyIntegrate(f, Sym(0), Sym(1)) = pi\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 7, | |
| "data": { | |
| "text/plain": "3.141592653589793" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "f(x) = 4/(1+x^2)\na, b = 0.0, 1.0\nN = 10\n\n@show RiemannSum(f, a, b, N)\n@show TrapezoidalRule(f, a, b, N)\n@show SimpsonRule(f, a, b, N)\n@show s = SymPyIntegrate(f, Sym(0), Sym(1))\nfloat(s)", | |
| "execution_count": 8, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "RiemannSum(f, a, b, N) = 3.142425985001098\nTrapezoidalRule(f, a, b, N) = 3.1399259889071587\nSimpsonRule(f, a, b, N) = 3.1415926529697855\ns = SymPyIntegrate(f, Sym(0), Sym(1)) = pi\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 8, | |
| "data": { | |
| "text/plain": "3.141592653589793" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "SimpsonRule(f, a, b, N) - π", | |
| "execution_count": 9, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 9, | |
| "data": { | |
| "text/plain": "-6.200076008155975e-10" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "$\\ds 4\\int_0^1\\frac{dx}{1+x^2}=\\pi$ の左辺を10分割のSimpsonの公式で計算すると, 小数点以下8桁まで円周率を正確に求めることができている！たったの10分割しかしていないのに！" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**注意:** <a href=\"https://www.google.co.jp/search?q=%22%E6%95%B0%E5%80%A4%E7%A9%8D%E5%88%86%22\">数値積分</a>は非常に深い分野である. 巧妙な幾つかの方法が開発されており, 多くの場合に以上で説明した方法よりもずっと効率的である. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "## 広義積分\n\n無限区間もしくは区間 $(a,b)$ の端で発散しているかもしれない函数 $f(x)$ について極限\n\n$$\n\\lim_{\\alpha\\searrow a,\\ \\beta\\nearrow b} \\int_\\alpha^\\beta f(x)\\,dx\n$$\n\nを**広義積分**と呼ぶ. 極限が存在しなければ広義積分は存在しない. 広義積分は\n\n$$\n\\lim_{\\alpha\\searrow a,\\ \\beta\\nearrow b} \\int_\\alpha^\\beta |f(x)|\\,dx\n\\tag{$*$}\n$$\n\nが有限の値に収束するとき収束し, そのときその広義積分は**絶対収束**するという. もとの広義積分は収束するが, ($*$)が有限の値に収束しないとき, その広義積分は**条件収束**するという.\n\n応用上重要な積分のかなりの割合が無限区間 $(-\\infty, \\infty)$, $(0,\\infty)$ 上の積分になり, 有限区間 $(a,b)$ 上の積分であっても広義積分になっていることが多い.\n\n例えば, **Gauss積分**や**Fourier変換**は無限区間 $(-\\infty,\\infty)$ 上の積分で定義され, **ガンマ函数**は半無限区間 $(0,\\infty)$ 上の積分で定義され, **ベータ函数**は有限開区間 $(0,1)$ 上の広義積分で定義される." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**注意:** 広義積分を表すときに毎回 $\\lim$ 記号を書くのは面倒なので, 通常の積分と同じ記号法で\n\n$$\n\\int_a^b f(x)\\,dx = \\lim_{\\alpha\\searrow a,\\ \\beta\\nearrow b} \\int_\\alpha^\\beta f(x)\\,dx\n$$\n\nと書くことが多い. さらに $f(x)=F'(x)$ のとき,\n\n$$\n\\lim_{\\alpha\\searrow a,\\ \\beta\\nearrow b} \\int_\\alpha^\\beta f(x)\\,dx =\n\\lim_{\\alpha\\searrow a,\\ \\beta\\nearrow b} [F(x)]_\\alpha^\\beta =\n\\lim_{\\beta\\nearrow b} F(\\beta) - \\lim_{\\alpha\\searrow a}F(\\alpha).\n$$\n\nであるが, これの右辺を記号の簡単のため\n\n$$\n[F(x)]_a^b = \\lim_{\\beta\\nearrow b} F(\\beta) - \\lim_{\\alpha\\searrow a}F(\\alpha)\n$$\n\nと書くことにする. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題:** 積分は積分の基本性質を満たしていると仮定する. $a<\\alpha\\leqq\\beta<b$ のとき積分 $\\int_\\alpha^\\beta f(x)\\,dx$ が定義されており, \n\n$$\n\\lim_{\\alpha\\searrow a,\\; \\beta\\nearrow b}\\int_\\alpha^\\beta |f(x)|\\,dx \n$$\n\nが有限の値に収束するならば,\n\n$$\n\\lim_{\\alpha\\searrow a,\\; \\beta\\nearrow b}\\int_\\alpha^\\beta f(x)\\,dx \n$$\n\nも収束することを示せ.\n\n**解答例:** $a$ に収束する単調減少数列 $a_n$ と $b$ に収束する単調増加数列 $b_n$ で $a_n\\leqq b_n$ を満たすものを任意に取る. このとき仮定より\n\n$$\n\\lim_{n\\to\\infty}\\int_{a_n}^{b_n} |f(x)|\\,dx \n$$\n\nは有限の値に収束する. この条件のもとで\n\n$$\n\\lim_{n\\to\\infty}\\int_{a_n}^{b_n} f(x)\\,dx \n$$\n\nが収束することを示せばよい. $A_n$, $B_n$ を\n\n$$\n\\begin{aligned}\n&\nA_1 = \\int_{a_1}^{b_1}|f(x)|\\,dx, & &\nA_n = \\int_{a_n}^{a_{n-1}}|f(x)|\\,dx + \\int_{b_{n-1}}^{b_n}|f(x)|\\,dx & & (n\\geqq 2)\n\\\\ &\nB_1 = \\int_{a_1}^{b_1}f(x)\\,dx, & &\nB_n = \\int_{a_n}^{a_{n-1}}f(x)\\,dx + \\int_{b_{n-1}}^{b_n}f(x)\\,dx & & (n\\geqq 2)\n\\end{aligned}\n$$\n\nと定める. このとき,\n\n$$\n|B_k|\\leqq A_k, \\quad\n\\int_{a_n}^{b_n} |f(x)|\\,dx = \\sum_{k=1}^n A_k, \\quad\n\\int_{a_n}^{b_n} f(x)\\,dx = \\sum_{k=1}^n B_k.\n$$\n\nそして, \n\n$$\n\\sum_{k=1}^\\infty |B_k| \\leqq \\sum_{k=1}^\\infty A_k= \\lim_{n\\to\\infty}\\int_{a_n}^{b_n} |f(x)|\\,dx < \\infty\n$$\n\nなので, 「級数の各項の絶対値を足し上げた結果が有限の値になればもとの級数は収束する」(所謂絶対収束)という結果より, 級数 $\\ds \\sum_{k=1}^\\infty B_k$ は収束する. それは $\\ds \\int_{a_n}^{b_n} f(x)\\,dx$ が $n\\to\\infty$ で収束することを意味する. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**注意:** 部分積分や置換積分は極限操作に注意を払えばそのまま広義積分にも適用可能である. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**注意:** 部分積分\n\n$$\n\\int_a^b f'(x)g(x)\\,dx = [f(x)g(x)]_a^b - \\int_a^b f(x)g'(x)\\,dx\n$$\n\nは, 特に $[f(x)g(x)]_a^b=0$ すなわち\n\n$$\n\\lim_{\\alpha\\searrow a}f(\\alpha)g(\\alpha) = \\lim_{\\beta\\nearrow b}f(\\beta)g(\\beta)\n$$\n\nのときに便利である. そのとき, 部分積分の公式は\n\n$$\n\\int_a^b f'(x)g(x)\\,dx = -\\int_a^b f(x)g'(x)\\,dx\n$$\n\nとなる. これは $f$ の微分を $g$ に移動すると $-1$ 倍になることを意味している. 部分積分はこの形で使われることが多い. \n\n特に, $[f(x)g(x)]_a^b=0$ の十分条件として,\n\n$$\n\\lim_{\\alpha\\searrow a}f(\\alpha)g(\\alpha) = 0, \\quad\n\\lim_{\\beta\\nearrow b}f(\\beta)g(\\beta) = 0\n$$\n\nという条件は非常によく出て来る. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### Fourier変換\n\n**問題(Fourier変換):** 函数 $f(x)$ は $C^1$ 級であり, $x\\to\\pm\\infty$ で $|f(x)|$ と $|f'(x)|$ は十分な速さで $0$ に近付くと仮定し, \n\n$$\n\\hat{f}(p) = \\int_{-\\infty}^\\infty e^{-ipx}f(x)\\,dx\n$$\n\nとおく. $\\hat{f}(p)$ は $f(x)$ のFourier変換と呼ばれる. このとき\n\n$$\n(xf)^\\wedge(p) = -i\\frac{\\d}{\\d p}\\hat{f}(p), \\quad\n\\left(-i\\frac{\\d}{\\d x}f\\right)^\\wedge(p) = p\\,\\hat{f}(p) \n$$\n\nが成立することを示せ. \n\n**解答例:** 前者は次のように示される:\n\n$$\n-i\\frac{\\d}{\\d p}\\hat{f}(p) =\n-i\\int_{-\\infty}^\\infty (-ix)e^{-ipx} f(x)\\,dx =\n\\int_{-\\infty}^\\infty e^{-ipx} xf(x)\\,dx = (xf)^\\wedge(p).\n$$\n\n後者を示そう. この場合にはすぐ上の注意のタイプの部分積分を利用できるので, \n\n$$\n\\left(-i\\frac{\\d}{\\d x}f\\right)^\\wedge(p) =\n-i\\int_{-\\infty}^\\infty e^{-ipx} \\frac{\\d}{\\d x}f(x)\\,dx =\ni\\int_{-\\infty}^\\infty (-ip)e^{-ipx} f(x)\\,dx = p\\hat{f}(p).\n$$\n\n2つ目の等号で部分積分を使った. $\\QED$\n\nこのようにFourier変換には $x$ をかける操作と微分する操作を交換する性質を持っている." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 絶対収束する広義積分の例\n\n以下は絶対収束する広義積分の例である.\n\n#### arcsin\n\n**(1)** $\\ds \\arcsin y = \\int_0^y \\frac{dt}{\\sqrt{1-t^2}}$ なので $\\ds \\int_{-1}^1 \\frac{dt}{\\sqrt{1-t^2}} = \\pi$. 積分区間 $(-1,1)$ の両端で被積分函数 $\\ds\\frac{1}{\\sqrt{1-t^2}}$ は発散している.\n\n#### arctan\n\n**(2)** $\\ds \\arctan a = \\int_0^a \\frac{dt}{1+t^2}$ なので $\\ds \\int_{-\\infty}^\\infty \\frac{dt}{1+t^2} = \\pi$. これは無限区間の積分になっている. \n\n以上の2つの $\\pi$ は単位半円周の長さを意味する." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### ガンマ函数\n\n**(3)** $s>0$ に対するガンマ函数の定義 $\\ds \\Gamma(s)=\\int_0^\\infty e^{-x}x^{s-1}\\,dx$ は無限区間の積分になっており, $0<s<1$ のとき $x\\searrow 0$ で被積分函数は発散している. \n\n$x\\to\\infty$ で $e^{-x}x^{s-1}$ は急速に $0$ に近付くので $\\ds \\int_1^\\infty e^{-x}x^{s-1}\\,dx$ はどんな $s$ の値に対しても収束している. \n\n$s\\ne 0$ のとき $\\ds \\int x^{s-1}\\,dx = \\frac{x^s}{s}$ なので $s>0$ ならば $s<1$ であっても $\\ds \\int_0^1 x^{s-1}\\,ds=\\frac{1}{s}$ (有限の値)となる. これより $s>0$ のとき $\\ds \\int_0^1 e^{-x}x^{s-1}\\,dx$ も収束することがわかる. ($x>0$ で $e^{-x}<1$ となることに注意せよ.)" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題:** $0<s$ のとき($s<1$ であっても), 積分 $\\int_0^1 e^{-x}x^{s-1}\\,dx$ が有限の値に収束することを示せ.\n\n**解答例:** $x>0$ のとき $e^{-x}<1$ なので, $s>0$ のとき, $x\\searrow 0$ で $x^s\\to 0$ となることより, \n\n$$\n\\int_0^1 e^{-x}x^{s-1}\\,dx \\leqq \\int_0^2 x^{s-1}\\,dx = \\left[\\frac{x^s}{s}\\right]_0^1 = \\frac{1}{s} < \\infty.\n\\qquad \\QED\n$$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題:** 任意の実数 $s$ に対して, 積分 $\\ds \\int_1^\\infty e^{-x}x^{s-1}\\,dx$ が有限の値に収束することを示せ.\n\n**解答例:** 指数函数は多項式函数よりも速く増大するので, $R>1$ を十分大きくすると, $x\\geqq R$ で $\\ds e^{-x/2}x^{s-1}\\leqq 1$ となる. ゆえに\n\n$$\ne^{-x}x^{s-1} \\leqq e^{-x/2}\\quad (x\\geqq R).\n$$\n\nゆえに\n\n$$\n\\begin{aligned}\n\\int_1^\\infty e^{-x}x^{s-1}\\,dx &= \n\\int_1^R e^{-x}x^{s-1}\\,dx + \\int_R^\\infty e^{-x}x^{s-1}\\,dx \n\\\\ &\\leqq\n\\int_1^R e^{-x}x^{s-1}\\,dx + \\int_R^\\infty e^{-x/2}\\,dx\n\\\\ &=\n\\int_1^R e^{-x}x^{s-1}\\,dx + 2e^{-R/2} < \\infty.\n\\qquad \\QED\n\\end{aligned}\n$$\n\n**注意:** 指数函数部分 $e^{-x}$ を $e^{-x/2}$ と $e^{-x/2}$ に分けてその片方をもう1つの因子と比較するというような方法はよく使われる. $\\QED$" | |
| }, | |
| { | |
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| "source": "#### ベータ函数\n\n**(4)** $p>0$, $q>0$ に対するベータ函数の定義 $\\ds B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ の被積分函数は, $0<p<1$ ならば積分区間の左端で, $0<q<1$ ならば積分区間の右端で発散している.\n\nしかし, $p>0$ ならば $p<1$ であっても $\\ds \\int_0^1 x^{p-1}\\,ds=\\frac{1}{p}$ (有限の値)となることから左端でも積分は収束する. 同様にして, $q>0$ ならば右端でも積分は収束する.\n\n#### Gauss分布のFourier変換\n\n**(5)** $a>0$ と仮定する. $f(x)=e^{-x^2/a}$ のFourier変換\n\n$$\n\\hat{f}(p) = \n\\int_{-\\infty}^\\infty e^{-ipx} f(x)\\,dx =\n\\int_{-\\infty}^\\infty e^{-x^2/a}\\cos(px)\\,dx\n$$\n\nの積分区間は無限区間である. $x\\to\\pm\\infty$ で $e^{-x^2/a}$ が急速に $0$ に近付くのでこの積分は絶対収束する." | |
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| "source": "#### 三角函数のLaplace変換\n\n**問題(三角函数のLaplace変換):** $s>0$ と仮定する. 次の公式を示せ:\n\n$$\n\\int_0^\\infty e^{-st}\\cos(pt)\\,dt = \\frac{s}{s^2+p^2}, \\quad\n\\int_0^\\infty e^{-st}\\sin(pt)\\,dt = \\frac{p}{s^2+p^2}.\n$$\n\n**解答例1:** $\\ds f(s) = \\int_0^\\infty e^{-st}\\cos(pt)\\,dt$ とおく. $\\ds \\frac{\\d}{\\d t}\\left(\\frac{e^{-st}}{-s}\\right)=e^{-st}$ を使った部分積分を2回行う.\n\n$$\n\\begin{aligned}\nf(s) &=\n\\int_0^\\infty e^{-st}\\cos(pt)\\,dt =\n\\left[\\frac{e^{-st}}{-s}\\cos(pt)\\right]_0^\\infty -\n\\int_0^\\infty \\frac{e^{-st}}{-s}\\;(-p\\sin(pt))\\,dt\n\\\\ &=\n\\frac{1}{s} -\n\\frac{p}{s}\\int_0^\\infty e^{-st}\\sin(pt)\\,dt =\n\\frac{1}{s} -\n\\frac{p}{s}\\left(\n\\left[\\frac{e^{-st}}{-s}\\sin(pt)\\right]_0^\\infty -\n\\int_0^\\infty \\frac{e^{-st}}{-s}\\;p\\cos(pt)\\,dt\n\\right)\n\\\\ &=\n\\frac{1}{s} -\n\\frac{p^2}{s^2}f(s).\n\\end{aligned}\n$$\n\nこれを $f(s)$ について解けば前者の公式が得られる.\n\n$\\ds g(s) = \\int_0^\\infty e^{-st}\\sin(pt)\\,dt$ とおく. 上と同様にして,\n$$\n\\begin{aligned}\ng(s) &=\n\\int_0^\\infty e^{-st}\\sin(pt)\\,dt =\n\\left[\\frac{e^{-st}}{-s}\\sin(pt)\\right]_0^\\infty -\n\\int_0^\\infty \\frac{e^{-st}}{-s}\\;p\\cos(pt)\\,dt\n\\\\ &=\n\\frac{p}{s}\\int_0^\\infty e^{-st}\\cos(pt)\\,dt =\n\\frac{p}{s}\\left(\n\\left[\\frac{e^{-st}}{-s}\\cos(pt)\\right]_0^\\infty -\n\\int_0^\\infty \\frac{e^{-st}}{-s}\\;(-p\\sin(pt))\\,dt\n\\right)\n\\\\ &=\n\\frac{p}{s}\\left(\n\\frac{1}{s} - \\frac{p}{s}g(s)\n\\right).\n\\end{aligned}\n$$\n\nこれを $g(s)$ について解けば後者の公式が得られる. $\\QED$" | |
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| "source": "**解答例2:** $e^{ipx}=\\cos(px)+i\\sin(px)$ を用いて一挙に計算する.\n\n$$\n\\int_0^\\infty e^{-st}e^{ipx}\\,dx =\n\\int_0^\\infty e^{-(s-ip)t}\\,dx =\n\\left[\\frac{e^{-(s-ip)t}}{-(s-ip)}\\right]_0^\\infty =\n\\frac{1}{s-ip} = \\frac{s+ip}{s^2+p^2}.\n$$\n\nこの結果の実部と虚部のそれぞれから示したい公式の前者と後者が得られる. $\\QED$\n\n部分積分を2回繰り返すよりも計算が圧倒的に平易になっている." | |
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| "source": "t, s, p = symbols(\"t s p\", positive=true)\n\nI1 = sympy.Integral(exp(-s*t)*cos(p*t), (t,0,oo))\nI2 = sympy.Integral(exp(-s*t)*sin(p*t), (t,0,oo))\n\nld(I1, \" = \", I1.doit().simplify())\nld(I2, \" = \", I2.doit().simplify())", | |
| "execution_count": 10, | |
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| "text/plain": "L\"$$\\int_{0}^{\\infty} e^{- s t} \\cos{\\left (p t \\right )}\\, dt = \\frac{s}{p^{2} + s^{2}}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} e^{- s t} \\cos{\\left (p t \\right )}\\, dt = \\frac{s}{p^{2} + s^{2}}$$" | |
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| "text/plain": "L\"$$\\int_{0}^{\\infty} e^{- s t} \\sin{\\left (p t \\right )}\\, dt = \\frac{p}{p^{2} + s^{2}}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} e^{- s t} \\sin{\\left (p t \\right )}\\, dt = \\frac{p}{p^{2} + s^{2}}$$" | |
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| "source": "### 条件収束する広義積分の例\n\n以下は条件収束する広義積分に関する有名な公式である. \n\n#### Dirichlet積分\n\n**(1)** **Dirichlet積分**(ディリクレ積分): $\\ds \\int_0^\\infty \\frac{\\sin x}{x}\\,dx = \\frac{\\pi}{2}$.\n\n#### Dirichlet積分の一般化\n\n**(2)** Dirichelet積分の一般化:\n\n$$\n\\int_0^\\infty \\frac{e^{-ax}\\sin x}{x}\\,dx = \n\\int_a^\\infty\\frac{dt}{1+t^2} = \\frac{\\pi}{2} - \\arctan a\n\\quad (a\\geqq 0).\n$$\n\nこれはDirichlet積分になる $a=0$ の場合にのみ条件収束する.\n\n#### Fresnel積分の特殊値\n\n**(3)** **Fresnel積分の特殊値**(フレネル積分の特殊値): \n\n$$\n\\int_{-\\infty}^\\infty \\cos(t^2)\\,dt = \n\\int_{-\\infty}^\\infty \\sin(t^2)\\,dt = \n\\sqrt{\\frac{\\pi}{2}}.\n$$\n\n**(4)** 上の続き. (3)における被積分函数は $t$ の偶函数なので $(0,\\infty)$ 上の積分に書き直して, $t=\\sqrt{x}$ と置換すると\n\n$$\n\\int_0^\\infty \\frac{\\cos x}{\\sqrt{x}}\\,dx = \n\\int_0^\\infty \\frac{\\sin x}{\\sqrt{x}}\\,dx = \n\\sqrt{\\frac{\\pi}{2}}.\n$$\n\nこれらの公式の証明については\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160501StirlingFormula.pdf\">ガンマ分布の中心極限定理とStirlingの公式</a>\n\nを参照せよ. (1), (2)の証明は第5.5節とを参照せよ. (1), (3), (4)の証明は第8.6節を参照せよ." | |
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| "source": "#### Dirichelt積分の収束の様子\n\n**問題:** $\\ds \\sinc x = \\frac{\\sin x}{x}$ ($x\\ne 0$), $\\sinc 0=1$ と $\\ds F(R) = \\int_0^R \\sinc x\\,dx$ のグラフをプロットせよ. $\\QED$\n\n次のセルを見よ." | |
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| "source": "f(x) = iszero(x) ? one(x) : sin(x)/x\nF(R) = quadgk(f, 0, R)[1]\nx = 0:0.05:100\nR = 0:0.05:100\n\nP1 = plot(x, f.(x), label=\"(sin x)/x\", legend=:topright, ylims=(-0.4, 1.2))\nhline!([0], ls=:dash, label=\"0\")\nP2 = plot(R, F.(R), label=\"int_0^R (sin x)/x dx\", legend=:bottomright, ylims=(0,2))\nhline!([π/2], ls=:dash, label=\"pi/2\")\nplot(P1, P2, size=(700,250))", | |
| "execution_count": 11, | |
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| "output_type": "execute_result", | |
| "execution_count": 11, | |
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217.217,154.24 217.776,155.767 218.335,157.726 218.893,160.115 219.452,162.929 220.011,166.165 220.57,169.818 \n 221.129,173.882 221.688,178.351 222.247,183.219 222.806,188.478 223.364,194.121 223.923,200.139 224.482,206.523 225.041,213.264 225.6,220.351 226.159,227.775 \n 226.718,235.524 227.276,243.586 227.835,251.951 228.394,260.606 228.953,269.538 229.512,278.734 230.071,288.181 230.63,297.865 231.189,307.772 231.747,317.888 \n 232.306,328.198 232.865,338.688 233.424,349.342 233.983,360.146 234.542,371.084 235.101,382.14 235.66,393.3 236.218,404.548 236.777,415.868 237.336,427.244 \n 237.895,438.661 238.454,450.103 239.013,461.555 239.572,473.002 240.13,484.427 240.689,495.817 241.248,507.155 241.807,518.427 242.366,529.619 242.925,540.716 \n 243.484,551.703 244.043,562.568 244.601,573.297 245.16,583.876 245.719,594.293 246.278,604.535 246.837,614.59 247.396,624.446 247.955,634.093 248.514,643.519 \n 249.072,652.714 249.631,661.669 250.19,670.373 250.749,678.819 251.308,686.997 251.867,694.9 252.426,702.52 252.984,709.851 253.543,716.886 254.102,723.621 \n 254.661,730.049 255.22,736.166 255.779,741.969 256.338,747.454 256.897,752.617 257.455,757.458 258.014,761.973 258.573,766.163 259.132,770.026 259.691,773.563 \n 260.25,776.775 260.809,779.661 261.367,782.225 261.926,784.468 262.485,786.394 263.044,788.005 263.603,789.305 264.162,790.299 264.721,790.992 265.28,791.388 \n 265.838,791.494 266.397,791.315 266.956,790.859 267.515,790.132 268.074,789.141 268.633,787.896 269.192,786.402 269.751,784.671 270.309,782.709 270.868,780.526 \n 271.427,778.132 271.986,775.536 272.545,772.748 273.104,769.778 273.663,766.636 274.221,763.334 274.78,759.881 275.339,756.289 275.898,752.568 276.457,748.728 \n 277.016,744.783 277.575,740.741 278.134,736.615 278.692,732.415 279.251,728.152 279.81,723.838 280.369,719.483 280.928,715.099 281.487,710.695 282.046,706.283 \n 282.605,701.873 283.163,697.475 283.722,693.099 284.281,688.756 284.84,684.455 285.399,680.205 285.958,676.016 286.517,671.896 287.075,667.854 287.634,663.898 \n 288.193,660.037 288.752,656.278 289.311,652.629 289.87,649.095 290.429,645.685 290.988,642.404 291.546,639.258 292.105,636.253 292.664,633.393 293.223,630.684 \n 293.782,628.13 294.341,625.734 294.9,623.5 295.459,621.431 296.017,619.53 296.576,617.798 297.135,616.239 297.694,614.852 298.253,613.64 298.812,612.602 \n 299.371,611.739 299.929,611.051 300.488,610.536 301.047,610.194 301.606,610.023 302.165,610.023 302.724,610.189 303.283,610.52 303.842,611.013 304.4,611.665 \n 304.959,612.473 305.518,613.431 306.077,614.537 306.636,615.785 307.195,617.171 307.754,618.69 308.313,620.336 308.871,622.105 309.43,623.99 309.989,625.985 \n 310.548,628.085 311.107,630.283 311.666,632.572 312.225,634.947 312.783,637.399 313.342,639.923 313.901,642.512 314.46,645.158 315.019,647.854 315.578,650.594 \n 316.137,653.37 316.696,656.175 317.254,659.002 317.813,661.844 318.372,664.694 318.931,667.545 319.49,670.389 320.049,673.221 320.608,676.033 321.167,678.819 \n 321.725,681.571 322.284,684.285 322.843,686.953 323.402,689.57 323.961,692.13 324.52,694.627 325.079,697.055 325.637,699.41 326.196,701.687 326.755,703.88 \n 327.314,705.986 327.873,708 328.432,709.917 328.991,711.735 329.55,713.45 330.108,715.058 330.667,716.557 331.226,717.945 331.785,719.218 332.344,720.375 \n 332.903,721.414 333.462,722.334 334.021,723.133 334.579,723.811 335.138,724.368 335.697,724.802 336.256,725.115 336.815,725.306 337.374,725.376 337.933,725.326 \n 338.491,725.157 339.05,724.871 339.609,724.47 340.168,723.955 340.727,723.329 341.286,722.594 341.845,721.753 342.404,720.81 342.962,719.766 343.521,718.627 \n 344.08,717.394 344.639,716.073 345.198,714.667 345.757,713.181 346.316,711.618 346.874,709.983 347.433,708.282 347.992,706.517 348.551,704.695 349.11,702.821 \n 349.669,700.899 350.228,698.934 350.787,696.932 351.345,694.898 351.904,692.837 352.463,690.754 353.022,688.656 353.581,686.546 354.14,684.43 354.699,682.315 \n 355.258,680.203 355.816,678.102 356.375,676.016 356.934,673.949 357.493,671.908 358.052,669.896 358.611,667.919 359.17,665.981 359.728,664.086 360.287,662.239 \n 360.846,660.444 361.405,658.705 361.964,657.026 362.523,655.411 363.082,653.862 363.641,652.384 364.199,650.979 364.758,649.651 365.317,648.402 365.876,647.234 \n 366.435,646.15 366.994,645.152 367.553,644.242 368.112,643.421 368.67,642.69 369.229,642.052 369.788,641.506 370.347,641.053 370.906,640.694 371.465,640.43 \n 372.024,640.259 372.582,640.182 373.141,640.199 373.7,640.307 374.259,640.508 374.818,640.799 375.377,641.179 375.936,641.647 376.495,642.201 377.053,642.839 \n 377.612,643.558 378.171,644.357 378.73,645.233 379.289,646.182 379.848,647.204 380.407,648.293 380.966,649.448 381.524,650.664 382.083,651.939 382.642,653.269 \n 383.201,654.651 383.76,656.08 384.319,657.552 384.878,659.064 385.436,660.613 385.995,662.192 386.554,663.8 387.113,665.431 387.672,667.081 388.231,668.746 \n 388.79,670.422 389.349,672.105 389.907,673.79 390.466,675.473 391.025,677.151 391.584,678.818 392.143,680.472 392.702,682.107 393.261,683.721 393.82,685.308 \n 394.378,686.866 394.937,688.391 395.496,689.879 396.055,691.327 396.614,692.731 397.173,694.089 397.732,695.396 398.29,696.651 398.849,697.85 399.408,698.991 \n 399.967,700.071 400.526,701.088 401.085,702.04 401.644,702.925 402.203,703.742 402.761,704.487 403.32,705.161 403.879,705.762 404.438,706.288 404.997,706.74 \n 405.556,707.115 406.115,707.415 406.674,707.638 407.232,707.784 407.791,707.854 408.35,707.848 408.909,707.765 409.468,707.608 410.027,707.376 410.586,707.071 \n 411.144,706.694 411.703,706.246 412.262,705.728 412.821,705.143 413.38,704.493 413.939,703.778 414.498,703.002 415.057,702.166 415.615,701.273 416.174,700.326 \n 416.733,699.327 417.292,698.279 417.851,697.184 418.41,696.047 418.969,694.869 419.528,693.654 420.086,692.405 420.645,691.126 421.204,689.82 421.763,688.489 \n 422.322,687.139 422.881,685.771 423.44,684.39 423.998,682.999 424.557,681.601 425.116,680.201 425.675,678.801 426.234,677.405 426.793,676.016 427.352,674.638 \n 427.911,673.275 428.469,671.928 429.028,670.603 429.587,669.301 430.146,668.027 430.705,666.782 431.264,665.571 431.823,664.395 432.381,663.258 432.94,662.162 \n 433.499,661.11 434.058,660.103 434.617,659.145 435.176,658.238 435.735,657.382 436.294,656.581 436.852,655.836 437.411,655.149 437.97,654.521 438.529,653.953 \n 439.088,653.446 439.647,653.001 440.206,652.62 440.765,652.302 441.323,652.048 441.882,651.859 442.441,651.735 443,651.675 443.559,651.68 444.118,651.749 \n 444.677,651.881 445.235,652.077 445.794,652.335 446.353,652.654 446.912,653.033 447.471,653.472 448.03,653.967 448.589,654.519 449.148,655.126 449.706,655.785 \n 450.265,656.495 450.824,657.254 451.383,658.059 451.942,658.909 452.501,659.801 453.06,660.733 453.619,661.702 454.177,662.705 454.736,663.741 455.295,664.806 \n 455.854,665.897 456.413,667.012 456.972,668.148 457.531,669.302 458.089,670.47 458.648,671.651 459.207,672.84 459.766,674.036 460.325,675.234 460.884,676.433 \n 461.443,677.628 462.002,678.818 462.56,679.999 463.119,681.167 463.678,682.322 464.237,683.458 464.796,684.575 465.355,685.669 465.914,686.737 466.473,687.778 \n 467.031,688.788 467.59,689.765 468.149,690.707 468.708,691.611 469.267,692.476 469.826,693.3 470.385,694.081 470.943,694.817 471.502,695.506 472.061,696.147 \n 472.62,696.738 473.179,697.279 473.738,697.768 474.297,698.204 474.856,698.586 475.414,698.914 475.973,699.187 476.532,699.404 477.091,699.565 477.65,699.671 \n 478.209,699.721 478.768,699.715 479.327,699.653 479.885,699.536 480.444,699.364 481.003,699.139 481.562,698.86 482.121,698.528 482.68,698.146 483.239,697.713 \n 483.797,697.232 484.356,696.703 484.915,696.128 485.474,695.509 486.033,694.848 486.592,694.146 487.151,693.404 487.71,692.626 488.268,691.814 488.827,690.968 \n 489.386,690.092 489.945,689.189 490.504,688.259 491.063,687.307 491.622,686.333 492.181,685.341 492.739,684.334 493.298,683.313 493.857,682.281 494.416,681.242 \n 494.975,680.197 495.534,679.15 496.093,678.102 496.651,677.057 497.21,676.017 497.769,674.984 498.328,673.961 498.887,672.952 499.446,671.957 500.005,670.98 \n 500.564,670.023 501.122,669.088 501.681,668.177 502.24,667.293 502.799,666.438 503.358,665.613 503.917,664.821 504.476,664.064 505.035,663.342 505.593,662.659 \n 506.152,662.015 506.711,661.412 507.27,660.851 507.829,660.334 508.388,659.861 508.947,659.434 509.505,659.053 510.064,658.72 510.623,658.435 511.182,658.198 \n 511.741,658.009 512.3,657.87 512.859,657.781 513.418,657.74 513.976,657.749 514.535,657.807 515.094,657.913 515.653,658.068 516.212,658.271 516.771,658.52 \n 517.33,658.816 517.889,659.157 518.447,659.542 519.006,659.971 519.565,660.441 520.124,660.952 520.683,661.503 521.242,662.091 521.801,662.715 522.359,663.373 \n 522.918,664.064 523.477,664.786 524.036,665.537 524.595,666.315 525.154,667.118 525.713,667.943 526.272,668.789 526.83,669.654 527.389,670.535 527.948,671.43 \n 528.507,672.336 529.066,673.252 529.625,674.176 530.184,675.104 530.742,676.034 531.301,676.965 531.86,677.893 532.419,678.817 532.978,679.735 533.537,680.643 \n 534.096,681.54 534.655,682.424 535.213,683.292 535.772,684.142 536.331,684.973 536.89,685.782 537.449,686.567 538.008,687.326 538.567,688.059 539.126,688.762 \n 539.684,689.434 540.243,690.075 540.802,690.681 541.361,691.252 541.92,691.787 542.479,692.284 543.038,692.742 543.596,693.161 544.155,693.539 544.714,693.875 \n 545.273,694.169 545.832,694.42 546.391,694.628 546.95,694.792 547.509,694.913 548.067,694.989 548.626,695.021 549.185,695.01 549.744,694.954 550.303,694.855 \n 550.862,694.712 551.421,694.527 551.98,694.3 552.538,694.032 553.097,693.722 553.656,693.374 554.215,692.986 554.774,692.561 555.333,692.1 555.892,691.603 \n 556.45,691.073 557.009,690.51 557.568,689.917 558.127,689.294 558.686,688.644 559.245,687.968 559.804,687.267 560.363,686.545 560.921,685.801 561.48,685.04 \n 562.039,684.262 562.598,683.469 563.157,682.663 563.716,681.848 564.275,681.023 564.834,680.193 565.392,679.358 565.951,678.521 566.51,677.683 567.069,676.848 \n 567.628,676.017 568.187,675.192 568.746,674.375 569.304,673.569 569.863,672.774 570.422,671.994 570.981,671.229 571.54,670.483 572.099,669.756 572.658,669.05 \n 573.217,668.368 573.775,667.71 574.334,667.079 574.893,666.475 575.452,665.9 576.011,665.356 576.57,664.844 577.129,664.365 577.688,663.92 578.246,663.509 \n 578.805,663.135 579.364,662.798 579.923,662.498 580.482,662.236 581.041,662.013 581.6,661.829 582.158,661.685 582.717,661.58 583.276,661.515 583.835,661.49 \n 584.394,661.505 584.953,661.559 585.512,661.653 586.071,661.786 586.629,661.958 587.188,662.167 587.747,662.415 588.306,662.698 588.865,663.018 589.424,663.373 \n 589.983,663.761 590.542,664.183 591.1,664.636 591.659,665.12 592.218,665.633 592.777,666.174 593.336,666.741 593.895,667.334 594.454,667.95 595.012,668.587 \n 595.571,669.245 596.13,669.921 596.689,670.614 597.248,671.322 597.807,672.044 598.366,672.776 598.925,673.518 599.483,674.267 600.042,675.022 600.601,675.781 \n 601.16,676.542 601.719,677.303 602.278,678.062 602.837,678.817 603.396,679.566 603.954,680.308 604.513,681.04 605.072,681.762 605.631,682.47 606.19,683.164 \n 606.749,683.842 607.308,684.501 607.866,685.141 608.425,685.76 608.984,686.356 609.543,686.928 610.102,687.475 610.661,687.996 611.22,688.488 611.779,688.951 \n 612.337,689.385 612.896,689.787 613.455,690.157 614.014,690.494 614.573,690.798 615.132,691.068 615.691,691.302 616.249,691.502 616.808,691.665 617.367,691.793 \n 617.926,691.884 618.485,691.94 619.044,691.958 619.603,691.94 620.162,691.887 620.72,691.797 621.279,691.671 621.838,691.51 622.397,691.315 622.956,691.085 \n 623.515,690.822 624.074,690.526 624.633,690.199 625.191,689.84 625.75,689.451 626.309,689.033 626.868,688.587 627.427,688.115 627.986,687.617 628.545,687.095 \n 629.103,686.551 629.662,685.985 630.221,685.399 630.78,684.795 631.339,684.174 631.898,683.538 632.457,682.888 633.016,682.227 633.574,681.555 634.133,680.874 \n 634.692,680.187 635.251,679.495 635.81,678.799 636.369,678.102 636.928,677.405 637.487,676.709 638.045,676.018 638.604,675.332 639.163,674.652 639.722,673.982 \n 640.281,673.321 640.84,672.673 641.399,672.038 641.957,671.419 642.516,670.816 643.075,670.231 643.634,669.665 644.193,669.12 644.752,668.598 645.311,668.098 \n 645.87,667.624 646.428,667.175 646.987,666.753 647.546,666.358 648.105,665.992 648.664,665.656 649.223,665.349 649.782,665.074 650.341,664.83 650.899,664.619 \n 651.458,664.439 652.017,664.293 652.576,664.18 653.135,664.1 653.694,664.053 654.253,664.04 654.811,664.061 655.37,664.114 655.929,664.201 656.488,664.321 \n 657.047,664.473 657.606,664.657 658.165,664.872 658.724,665.118 659.282,665.394 659.841,665.699 660.4,666.033 660.959,666.395 661.518,666.783 662.077,667.197 \n 662.636,667.635 663.195,668.096 663.753,668.58 664.312,669.084 664.871,669.608 665.43,670.151 665.989,670.71 666.548,671.284 667.107,671.872 667.665,672.473 \n 668.224,673.084 668.783,673.705 669.342,674.334 669.901,674.968 670.46,675.608 671.019,676.25 671.578,676.893 672.136,677.537 672.695,678.178 673.254,678.816 \n 673.813,679.449 674.372,680.075 674.931,680.693 675.49,681.302 676.049,681.899 676.607,682.483 677.166,683.054 677.725,683.609 678.284,684.147 678.843,684.668 \n 679.402,685.168 679.961,685.649 680.519,686.107 681.078,686.543 681.637,686.955 682.196,687.342 682.755,687.704 683.314,688.039 683.873,688.346 684.432,688.626 \n 684.99,688.877 685.549,689.098 686.108,689.29 686.667,689.452 687.226,689.584 687.785,689.685 688.344,689.755 688.903,689.794 689.461,689.802 690.02,689.779 \n 690.579,689.725 691.138,689.641 691.697,689.526 692.256,689.381 692.815,689.207 693.373,689.004 693.932,688.772 694.491,688.513 695.05,688.226 695.609,687.914 \n 696.168,687.575 696.727,687.212 697.286,686.826 697.844,686.417 698.403,685.986 698.962,685.535 699.521,685.065 700.08,684.576 700.639,684.071 701.198,683.551 \n 701.756,683.016 702.315,682.469 702.874,681.91 703.433,681.342 703.992,680.765 704.551,680.18 705.11,679.591 705.669,678.997 706.227,678.4 706.786,677.803 \n 707.345,677.206 707.904,676.611 708.463,676.019 709.022,675.432 709.581,674.851 710.14,674.278 710.698,673.714 711.257,673.161 711.816,672.619 712.375,672.091 \n 712.934,671.577 713.493,671.079 714.052,670.598 714.61,670.135 715.169,669.691 715.728,669.267 716.287,668.865 716.846,668.485 717.405,668.128 717.964,667.795 \n 718.523,667.487 719.081,667.204 719.64,666.948 720.199,666.718 720.758,666.516 721.317,666.341 721.876,666.194 722.435,666.075 722.994,665.985 723.552,665.924 \n 724.111,665.892 724.67,665.888 725.229,665.913 725.788,665.967 726.347,666.05 726.906,666.16 727.464,666.299 728.023,666.465 728.582,666.658 729.141,666.877 \n 729.7,667.122 730.259,667.392 730.818,667.687 731.377,668.006 731.935,668.347 732.494,668.71 733.053,669.094 733.612,669.498 734.171,669.921 734.73,670.362 \n 735.289,670.819 735.848,671.292 736.406,671.78 736.965,672.28 737.524,672.792 738.083,673.314 738.642,673.846 739.201,674.385 739.76,674.931 740.318,675.482 \n 740.877,676.036 741.436,676.593 741.995,677.151 742.554,677.708 743.113,678.263 743.672,678.815 744.231,679.362 744.789,679.903 745.348,680.437 745.907,680.963 \n 746.466,681.478 747.025,681.982 747.584,682.474 748.143,682.951 748.702,683.415 749.26,683.862 749.819,684.292 750.378,684.704 750.937,685.097 751.496,685.47 \n 752.055,685.822 752.614,686.153 753.172,686.461 753.731,686.745 754.29,687.006 754.849,687.242 755.408,687.454 755.967,687.64 756.526,687.799 757.085,687.933 \n 757.643,688.04 758.202,688.121 758.761,688.174 759.32,688.201 759.879,688.2 760.438,688.173 760.997,688.119 761.556,688.038 762.114,687.931 762.673,687.797 \n 763.232,687.639 763.791,687.455 764.35,687.246 764.909,687.013 765.468,686.757 766.026,686.478 766.585,686.176 767.144,685.854 767.703,685.511 768.262,685.149 \n 768.821,684.768 769.38,684.369 769.939,683.954 770.497,683.524 771.056,683.079 771.615,682.621 772.174,682.15 772.733,681.669 773.292,681.178 773.851,680.679 \n 774.41,680.173 774.968,679.661 775.527,679.144 776.086,678.624 776.645,678.102 777.204,677.579 777.763,677.057 778.322,676.537 778.88,676.02 779.439,675.507 \n 779.998,675.001 780.557,674.501 781.116,674.01 781.675,673.528 782.234,673.057 782.793,672.597 783.351,672.151 783.91,671.718 784.469,671.301 785.028,670.9 \n 785.587,670.516 786.146,670.15 786.705,669.802 787.263,669.475 787.822,669.167 788.381,668.882 788.94,668.617 789.499,668.376 790.058,668.157 790.617,667.962 \n 791.176,667.791 791.734,667.645 792.293,667.523 792.852,667.426 793.411,667.354 793.97,667.307 794.529,667.286 795.088,667.29 795.647,667.319 796.205,667.374 \n 796.764,667.453 797.323,667.557 797.882,667.686 798.441,667.839 799,668.015 799.559,668.214 800.117,668.436 800.676,668.68 801.235,668.946 801.794,669.232 \n 802.353,669.538 802.912,669.863 803.471,670.206 804.03,670.567 804.588,670.944 805.147,671.336 805.706,671.743 806.265,672.164 806.824,672.596 807.383,673.04 \n 807.942,673.494 808.501,673.957 809.059,674.428 809.618,674.905 810.177,675.388 810.736,675.874 811.295,676.364 811.854,676.856 812.413,677.347 812.971,677.839 \n 813.53,678.328 814.089,678.814 814.648,679.296 815.207,679.772 815.766,680.241 816.325,680.702 816.884,681.154 817.442,681.597 818.001,682.027 818.56,682.446 \n 819.119,682.851 819.678,683.242 820.237,683.618 820.796,683.977 821.355,684.32 821.913,684.644 822.472,684.95 823.031,685.237 823.59,685.503 824.149,685.749 \n 824.708,685.973 825.267,686.176 825.825,686.357 826.384,686.514 826.943,686.649 827.502,686.761 828.061,686.849 828.62,686.913 829.179,686.954 829.738,686.97 \n 830.296,686.963 830.855,686.932 831.414,686.877 831.973,686.798 832.532,686.697 833.091,686.572 833.65,686.425 834.209,686.255 834.767,686.064 835.326,685.852 \n 835.885,685.618 836.444,685.365 837.003,685.092 837.562,684.801 838.121,684.492 838.679,684.166 839.238,683.823 839.797,683.465 840.356,683.092 840.915,682.706 \n 841.474,682.308 842.033,681.898 842.592,681.477 843.15,681.048 843.709,680.609 844.268,680.164 844.827,679.713 845.386,679.256 845.945,678.796 846.504,678.333 \n 847.063,677.869 847.621,677.405 848.18,676.941 848.739,676.479 849.298,676.021 849.857,675.567 850.416,675.118 850.975,674.675 851.533,674.241 852.092,673.815 \n 852.651,673.399 853.21,672.993 853.769,672.6 854.328,672.219 854.887,671.851 855.446,671.498 856.004,671.161 856.563,670.84 857.122,670.536 857.681,670.249 \n 858.24,669.981 858.799,669.732 859.358,669.503 859.917,669.293 860.475,669.105 861.034,668.937 861.593,668.791 862.152,668.667 862.711,668.565 863.27,668.485 \n 863.829,668.427 864.387,668.392 864.946,668.38 865.505,668.39 866.064,668.423 866.623,668.478 867.182,668.556 867.741,668.655 868.3,668.776 868.858,668.919 \n 869.417,669.082 869.976,669.266 870.535,669.47 871.094,669.694 871.653,669.937 872.212,670.197 872.77,670.476 873.329,670.771 873.888,671.082 874.447,671.409 \n 875.006,671.75 875.565,672.104 876.124,672.471 876.683,672.85 877.241,673.24 877.8,673.64 878.359,674.048 878.918,674.464 879.477,674.887 880.036,675.315 \n 880.595,675.748 881.154,676.184 881.712,676.623 882.271,677.063 882.83,677.503 883.389,677.942 883.948,678.379 884.507,678.813 885.066,679.242 885.624,679.667 \n 886.183,680.085 886.742,680.496 887.301,680.898 887.86,681.291 888.419,681.674 888.978,682.045 889.537,682.404 890.095,682.75 890.654,683.083 891.213,683.4 \n 891.772,683.702 892.331,683.988 892.89,684.257 893.449,684.509 894.008,684.742 894.566,684.957 895.125,685.152 895.684,685.328 896.243,685.484 896.802,685.62 \n 897.361,685.734 897.92,685.828 898.478,685.901 899.037,685.952 899.596,685.982 900.155,685.99 900.714,685.977 901.273,685.943 901.832,685.887 902.391,685.811 \n 902.949,685.713 903.508,685.595 904.067,685.457 904.626,685.299 905.185,685.121 905.744,684.925 906.303,684.71 906.862,684.477 907.42,684.227 907.979,683.961 \n 908.538,683.678 909.097,683.38 909.656,683.068 910.215,682.742 910.774,682.404 911.332,682.053 911.891,681.692 912.45,681.32 913.009,680.94 913.568,680.551 \n 914.127,680.155 914.686,679.752 915.245,679.345 915.803,678.933 916.362,678.518 916.921,678.101 917.48,677.683 918.039,677.265 918.598,676.849 919.157,676.434 \n 919.716,676.022 920.274,675.614 920.833,675.212 921.392,674.816 921.951,674.427 922.51,674.046 923.069,673.674 923.628,673.311 924.186,672.96 924.745,672.621 \n 925.304,672.294 925.863,671.98 926.422,671.68 926.981,671.395 927.54,671.126 928.099,670.873 928.657,670.636 929.216,670.417 929.775,670.215 930.334,670.032 \n 930.893,669.868 931.452,669.723 932.011,669.597 932.57,669.491 933.128,669.405 933.687,669.339 934.246,669.293 934.805,669.268 935.364,669.263 935.923,669.278 \n 936.482,669.314 937.04,669.37 937.599,669.446 938.158,669.541 938.717,669.657 939.276,669.791 939.835,669.944 940.394,670.116 940.953,670.306 941.511,670.513 \n 942.07,670.737 942.629,670.977 943.188,671.234 943.747,671.505 944.306,671.79 944.865,672.089 945.424,672.401 945.982,672.725 946.541,673.061 947.1,673.406 \n 947.659,673.761 948.218,674.125 948.777,674.497 949.336,674.875 949.894,675.259 950.453,675.647 951.012,676.04 951.571,676.435 952.13,676.832 952.689,677.23 \n 953.248,677.628 953.807,678.025 954.365,678.42 954.924,678.811 955.483,679.199 956.042,679.581 956.601,679.958 957.16,680.327 957.719,680.689 958.277,681.042 \n 958.836,681.386 959.395,681.719 959.954,682.041 960.513,682.35 961.072,682.647 961.631,682.931 962.19,683.2 962.748,683.454 963.307,683.693 963.866,683.916 \n 964.425,684.123 964.984,684.312 965.543,684.484 966.102,684.638 966.661,684.773 967.219,684.89 967.778,684.989 968.337,685.068 968.896,685.128 969.455,685.168 \n 970.014,685.189 970.573,685.191 971.131,685.173 971.69,685.136 972.249,685.08 972.808,685.005 973.367,684.911 973.926,684.798 974.485,684.667 975.044,684.518 \n 975.602,684.352 976.161,684.168 976.72,683.968 977.279,683.752 977.838,683.52 978.397,683.273 978.956,683.013 979.515,682.738 980.073,682.451 980.632,682.151 \n 981.191,681.84 981.75,681.519 982.309,681.188 982.868,680.848 983.427,680.499 983.985,680.144 984.544,679.782 985.103,679.415 985.662,679.043 986.221,678.668 \n 986.78,678.291 987.339,677.911 987.898,677.531 988.456,677.152 989.015,676.773 989.574,676.397 990.133,676.023 990.692,675.654 991.251,675.29 991.81,674.931 \n 992.369,674.58 992.927,674.235 993.486,673.9 994.045,673.573 994.604,673.257 995.163,672.951 995.722,672.657 996.281,672.376 996.839,672.107 997.398,671.852 \n 997.957,671.611 998.516,671.385 999.075,671.175 999.634,670.98 1000.19,670.802 1000.75,670.641 1001.31,670.496 1001.87,670.369 1002.43,670.26 1002.99,670.169 \n 1003.55,670.096 1004.1,670.042 1004.66,670.006 1005.22,669.989 1005.78,669.99 1006.34,670.01 1006.9,670.048 1007.46,670.105 1008.02,670.179 1008.58,670.272 \n 1009.13,670.383 1009.69,670.511 1010.25,670.656 1010.81,670.817 1011.37,670.995 1011.93,671.189 1012.49,671.398 1013.05,671.621 1013.61,671.859 1014.16,672.111 \n 1014.72,672.375 1015.28,672.652 1015.84,672.94 1016.4,673.239 1016.96,673.547 1017.52,673.866 1018.08,674.192 1018.64,674.526 1019.19,674.867 1019.75,675.214 \n 1020.31,675.566 1020.87,675.922 1021.43,676.281 1021.99,676.642 1022.55,677.005 1023.11,677.369 1023.67,677.732 1024.22,678.094 1024.78,678.453 1025.34,678.81 \n 1025.9,679.163 1026.46,679.51 1027.02,679.852 1027.58,680.188 1028.14,680.516 1028.7,680.836 1029.25,681.147 1029.81,681.448 1030.37,681.738 1030.93,682.018 \n 1031.49,682.286 1032.05,682.541 1032.61,682.783 1033.17,683.011 1033.72,683.225 1034.28,683.424 1034.84,683.608 1035.4,683.776 1035.96,683.928 1036.52,684.064 \n 1037.08,684.182 1037.64,684.284 1038.2,684.368 1038.75,684.435 1039.31,684.485 1039.87,684.516 1040.43,684.53 1040.99,684.526 1041.55,684.505 1042.11,684.465 \n 1042.67,684.408 1043.23,684.334 1043.78,684.242 1044.34,684.134 1044.9,684.009 1045.46,683.868 1046.02,683.71 1046.58,683.538 1047.14,683.35 1047.7,683.147 \n 1048.26,682.931 1048.81,682.701 1049.37,682.458 1049.93,682.203 1050.49,681.936 1051.05,681.659 1051.61,681.371 1052.17,681.074 1052.73,680.768 1053.29,680.453 \n 1053.84,680.132 1054.4,679.805 1054.96,679.472 1055.52,679.134 1056.08,678.792 1056.64,678.448 1057.2,678.101 1057.76,677.753 1058.32,677.405 1058.87,677.057 \n 1059.43,676.71 1059.99,676.366 1060.55,676.025 1061.11,675.688 1061.67,675.355 1062.23,675.028 1062.79,674.708 1063.34,674.395 1063.9,674.089 1064.46,673.793 \n 1065.02,673.505 1065.58,673.228 1066.14,672.962 1066.7,672.707 1067.26,672.465 1067.82,672.235 1068.37,672.018 1068.93,671.815 1069.49,671.627 1070.05,671.453 \n 1070.61,671.294 1071.17,671.15 1071.73,671.023 1072.29,670.912 1072.85,670.817 1073.4,670.738 1073.96,670.677 1074.52,670.632 1075.08,670.604 1075.64,670.594 \n 1076.2,670.6 1076.76,670.624 1077.32,670.664 1077.88,670.721 1078.43,670.795 1078.99,670.886 1079.55,670.992 1080.11,671.115 1080.67,671.253 1081.23,671.406 \n 1081.79,671.574 1082.35,671.756 1082.91,671.953 1083.46,672.163 1084.02,672.385 1084.58,672.62 1085.14,672.867 1085.7,673.124 1086.26,673.392 1086.82,673.67 \n 1087.38,673.957 1087.94,674.252 1088.49,674.554 1089.05,674.863 1089.61,675.179 1090.17,675.499 1090.73,675.824 1091.29,676.152 1091.85,676.484 1092.41,676.817 \n 1092.96,677.151 1093.52,677.485 1094.08,677.819 1094.64,678.151 1095.2,678.481 1095.76,678.808 1096.32,679.132 1096.88,679.45 1097.44,679.763 1097.99,680.07 \n 1098.55,680.369 1099.11,680.661 1099.67,680.945 1100.23,681.219 1100.79,681.483 1101.35,681.737 1101.91,681.98 1102.47,682.211 1103.02,682.43 1103.58,682.636 \n 1104.14,682.829 1104.7,683.008 1105.26,683.173 1105.82,683.323 1106.38,683.458 1106.94,683.578 1107.5,683.683 1108.05,683.772 1108.61,683.844 1109.17,683.901 \n 1109.73,683.941 1110.29,683.965 1110.85,683.973 1111.41,683.964 1111.97,683.939 1112.53,683.897 1113.08,683.84 1113.64,683.766 1114.2,683.677 1114.76,683.572 \n 1115.32,683.452 1115.88,683.317 1116.44,683.168 1117,683.004 1117.56,682.827 1118.11,682.636 1118.67,682.432 1119.23,682.216 1119.79,681.989 1120.35,681.75 \n 1120.91,681.501 1121.47,681.242 1122.03,680.973 1122.59,680.696 1123.14,680.412 1123.7,680.12 1124.26,679.821 1124.82,679.517 1125.38,679.209 1125.94,678.896 \n 1126.5,678.58 1127.06,678.261 1127.61,677.94 1128.17,677.619 1128.73,677.298 1129.29,676.977 1129.85,676.658 1130.41,676.341 1130.97,676.027 1131.53,675.717 \n 1132.09,675.411 1132.64,675.111 1133.2,674.817 1133.76,674.53 1134.32,674.251 1134.88,673.979 1135.44,673.717 1136,673.464 1136.56,673.221 1137.12,672.99 \n 1137.67,672.769 1138.23,672.561 1138.79,672.364 1139.35,672.181 1139.91,672.011 1140.47,671.855 1141.03,671.712 1141.59,671.584 1142.15,671.471 1142.7,671.373 \n 1143.26,671.29 1143.82,671.223 1144.38,671.171 1144.94,671.134 1145.5,671.114 1146.06,671.109 1146.62,671.12 1147.18,671.147 1147.73,671.189 1148.29,671.247 \n 1148.85,671.32 1149.41,671.408 1149.97,671.512 1150.53,671.629 1151.09,671.762 1151.65,671.908 1152.21,672.067 1152.76,672.24 1153.32,672.426 1153.88,672.624 \n 1154.44,672.833 1155,673.054 1155.56,673.286 1156.12,673.527 1156.68,673.778 1157.23,674.038 1157.79,674.306 1158.35,674.581 1158.91,674.863 1159.47,675.151 \n 1160.03,675.444 1160.59,675.742 1161.15,676.044 1161.71,676.349 1162.26,676.656 1162.82,676.965 1163.38,677.275 1163.94,677.584 1164.5,677.893 1165.06,678.2 \n 1165.62,678.505 1166.18,678.807 1166.74,679.105 1167.29,679.398 1167.85,679.686 1168.41,679.968 1168.97,680.244 1169.53,680.512 1170.09,680.772 1170.65,681.023 \n 1171.21,681.265 1171.77,681.497 1172.32,681.718 1172.88,681.929 1173.44,682.128 1174,682.315 1174.56,682.49 1175.12,682.652 1175.68,682.8 1176.24,682.935 \n 1176.8,683.056 1177.35,683.163 1177.91,683.255 1178.47,683.333 1179.03,683.395 1179.59,683.443 1180.15,683.475 1180.71,683.493 1181.27,683.495 1181.83,683.482 \n 1182.38,683.454 1182.94,683.411 1183.5,683.353 1184.06,683.28 1184.62,683.192 1185.18,683.091 1185.74,682.975 1186.3,682.845 1186.86,682.702 1187.41,682.546 \n 1187.97,682.378 1188.53,682.197 1189.09,682.004 1189.65,681.8 1190.21,681.586 1190.77,681.361 1191.33,681.127 1191.88,680.884 1192.44,680.632 1193,680.373 \n 1193.56,680.106 1194.12,679.833 1194.68,679.555 1195.24,679.271 1195.8,678.983 1196.36,678.691 1196.91,678.397 1197.47,678.101 1198.03,677.803 1198.59,677.504 \n 1199.15,677.206 1199.71,676.908 1200.27,676.612 1200.83,676.319 1201.39,676.028 1201.94,675.742 1202.5,675.46 1203.06,675.183 1203.62,674.911 1204.18,674.647 \n 1204.74,674.39 1205.3,674.14 1205.86,673.899 1206.42,673.667 1206.97,673.445 1207.53,673.233 1208.09,673.032 1208.65,672.841 1209.21,672.663 1209.77,672.496 \n 1210.33,672.342 1210.89,672.201 1211.45,672.073 1212,671.959 1212.56,671.858 1213.12,671.771 1213.68,671.699 1214.24,671.64 1214.8,671.597 1215.36,671.568 \n 1215.92,671.553 1216.48,671.553 1217.03,671.568 1217.59,671.598 1218.15,671.642 1218.71,671.7 1219.27,671.773 1219.83,671.859 1220.39,671.96 1220.95,672.074 \n 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1658.49,244.172 1659.05,247.926 1659.61,251.558 1660.17,255.062 1660.73,258.43 1661.28,261.657 1661.84,264.738 1662.4,267.667 \n 1662.96,270.441 1663.52,273.055 1664.08,275.506 1664.64,277.79 1665.2,279.905 1665.76,281.848 1666.31,283.618 1666.87,285.213 1667.43,286.632 1667.99,287.875 \n 1668.55,288.942 1669.11,289.832 1669.67,290.547 1670.23,291.087 1670.78,291.454 1671.34,291.651 1671.9,291.678 1672.46,291.539 1673.02,291.237 1673.58,290.775 \n 1674.14,290.157 1674.7,289.386 1675.26,288.468 1675.81,287.405 1676.37,286.203 1676.93,284.868 1677.49,283.404 1678.05,281.817 1678.61,280.113 1679.17,278.297 \n 1679.73,276.376 1680.29,274.356 1680.84,272.244 1681.4,270.045 1681.96,267.767 1682.52,265.417 1683.08,263 1683.64,260.525 1684.2,257.997 1684.76,255.425 \n 1685.32,252.814 1685.87,250.173 1686.43,247.507 1686.99,244.825 1687.55,242.132 1688.11,239.435 1688.67,236.742 1689.23,234.059 1689.79,231.393 1690.35,228.749 \n 1690.9,226.135 1691.46,223.556 1692.02,221.018 1692.58,218.527 1693.14,216.089 1693.7,213.709 1694.26,211.393 1694.82,209.144 1695.38,206.969 1695.93,204.872 \n 1696.49,202.856 1697.05,200.926 1697.61,199.087 1698.17,197.34 1698.73,195.69 1699.29,194.139 1699.85,192.691 1700.41,191.348 1700.96,190.111 1701.52,188.983 \n 1702.08,187.966 1702.64,187.06 1703.2,186.267 1703.76,185.587 1704.32,185.021 1704.88,184.569 1705.43,184.231 1705.99,184.007 1706.55,183.895 1707.11,183.896 \n 1707.67,184.007 1708.23,184.228 1708.79,184.556 1709.35,184.99 1709.91,185.527 1710.46,186.166 1711.02,186.903 1711.58,187.736 1712.14,188.662 1712.7,189.677 \n 1713.26,190.778 1713.82,191.961 1714.38,193.223 1714.94,194.56 1715.49,195.967 1716.05,197.441 1716.61,198.977 1717.17,200.57 1717.73,202.217 1718.29,203.913 \n 1718.85,205.652 1719.41,207.431 1719.97,209.244 1720.52,211.087 1721.08,212.954 1721.64,214.841 1722.2,216.743 1722.76,218.655 1723.32,220.573 1723.88,222.491 \n 1724.44,224.404 1725,226.308 1725.55,228.199 1726.11,230.071 1726.67,231.92 1727.23,233.742 1727.79,235.533 1728.35,237.288 1728.91,239.003 1729.47,240.674 \n 1730.03,242.298 1730.58,243.871 1731.14,245.39 1731.7,246.85 1732.26,248.25 1732.82,249.586 1733.38,250.854 1733.94,252.054 1734.5,253.182 1735.05,254.235 \n 1735.61,255.213 1736.17,256.113 1736.73,256.934 1737.29,257.674 1737.85,258.332 1738.41,258.907 1738.97,259.399 1739.53,259.806 1740.08,260.129 1740.64,260.367 \n 1741.2,260.521 1741.76,260.59 1742.32,260.576 1742.88,260.479 1743.44,260.299 1744,260.038 1744.56,259.698 1745.11,259.279 1745.67,258.783 1746.23,258.213 \n 1746.79,257.57 1747.35,256.856 1747.91,256.074 1748.47,255.226 1749.03,254.314 1749.59,253.342 1750.14,252.312 1750.7,251.228 1751.26,250.092 1751.82,248.908 \n 1752.38,247.679 1752.94,246.408 1753.5,245.099 1754.06,243.755 1754.62,242.38 1755.17,240.978 1755.73,239.552 1756.29,238.106 1756.85,236.644 1757.41,235.17 \n 1757.97,233.687 1758.53,232.198 1759.09,230.709 1759.65,229.222 1760.2,227.742 1760.76,226.271 1761.32,224.813 1761.88,223.373 1762.44,221.953 1763,220.557 \n 1763.56,219.188 1764.12,217.849 1764.68,216.544 1765.23,215.275 1765.79,214.046 1766.35,212.859 1766.91,211.717 1767.47,210.622 1768.03,209.578 1768.59,208.585 \n 1769.15,207.646 1769.7,206.764 1770.26,205.94 1770.82,205.176 1771.38,204.472 1771.94,203.832 1772.5,203.255 1773.06,202.743 1773.62,202.296 1774.18,201.916 \n 1774.73,201.602 1775.29,201.356 1775.85,201.178 1776.41,201.066 1776.97,201.022 1777.53,201.045 1778.09,201.134 1778.65,201.289 1779.21,201.509 1779.76,201.793 \n 1780.32,202.14 1780.88,202.548 1781.44,203.017 1782,203.545 1782.56,204.129 1783.12,204.769 1783.68,205.462 1784.24,206.207 1784.79,207 1785.35,207.841 \n 1785.91,208.725 1786.47,209.652 1787.03,210.618 1787.59,211.621 1788.15,212.658 1788.71,213.726 1789.27,214.823 1789.82,215.945 1790.38,217.089 1790.94,218.253 \n 1791.5,219.434 1792.06,220.628 1792.62,221.833 1793.18,223.045 1793.74,224.261 1794.3,225.479 1794.85,226.695 1795.41,227.906 1795.97,229.109 1796.53,230.301 \n 1797.09,231.48 1797.65,232.643 1798.21,233.786 1798.77,234.907 1799.32,236.003 1799.88,237.072 1800.44,238.111 1801,239.118 1801.56,240.09 1802.12,241.026 \n 1802.68,241.922 1803.24,242.778 1803.8,243.59 1804.35,244.359 1804.91,245.08 1805.47,245.754 1806.03,246.379 1806.59,246.953 1807.15,247.475 1807.71,247.945 \n 1808.27,248.361 1808.83,248.722 1809.38,249.029 1809.94,249.28 1810.5,249.475 1811.06,249.615 1811.62,249.698 1812.18,249.725 1812.74,249.697 1813.3,249.613 \n 1813.86,249.475 1814.41,249.282 1814.97,249.036 1815.53,248.737 1816.09,248.387 1816.65,247.987 1817.21,247.537 1817.77,247.039 1818.33,246.496 1818.89,245.907 \n 1819.44,245.276 1820,244.603 1820.56,243.891 1821.12,243.142 1821.68,242.358 1822.24,241.54 1822.8,240.692 1823.36,239.815 1823.92,238.911 1824.47,237.984 \n 1825.03,237.035 1825.59,236.067 1826.15,235.082 1826.71,234.084 1827.27,233.074 1827.83,232.055 1828.39,231.03 1828.94,230.001 1829.5,228.972 1830.06,227.943 \n 1830.62,226.919 1831.18,225.901 1831.74,224.893 1832.3,223.896 1832.86,222.912 1833.42,221.946 1833.97,220.998 1834.53,220.07 1835.09,219.166 1835.65,218.288 \n 1836.21,217.437 1836.77,216.615 1837.33,215.824 1837.89,215.067 1838.45,214.344 1839,213.658 1839.56,213.01 1840.12,212.401 1840.68,211.833 1841.24,211.307 \n 1841.8,210.825 1842.36,210.386 1842.92,209.993 1843.48,209.645 1844.03,209.344 1844.59,209.089 1845.15,208.882 1845.71,208.723 1846.27,208.612 1846.83,208.549 \n 1847.39,208.533 1847.95,208.565 1848.51,208.645 1849.06,208.772 1849.62,208.945 1850.18,209.164 1850.74,209.428 1851.3,209.736 1851.86,210.088 1852.42,210.482 \n 1852.98,210.916 1853.54,211.391 1854.09,211.904 1854.65,212.453 1855.21,213.039 1855.77,213.658 1856.33,214.309 1856.89,214.99 1857.45,215.7 1858.01,216.437 \n 1858.57,217.198 1859.12,217.982 1859.68,218.786 1860.24,219.609 1860.8,220.448 1861.36,221.302 1861.92,222.167 1862.48,223.043 1863.04,223.925 1863.59,224.814 \n 1864.15,225.705 1864.71,226.597 1865.27,227.488 1865.83,228.375 1866.39,229.257 1866.95,230.13 1867.51,230.994 1868.07,231.845 1868.62,232.682 1869.18,233.503 \n 1869.74,234.305 1870.3,235.087 1870.86,235.847 1871.42,236.584 1871.98,237.294 1872.54,237.978 1873.1,238.632 1873.65,239.256 1874.21,239.848 1874.77,240.407 \n 1875.33,240.932 1875.89,241.421 1876.45,241.873 1877.01,242.288 1877.57,242.664 1878.13,243.001 1878.68,243.298 1879.24,243.554 1879.8,243.77 1880.36,243.943 \n 1880.92,244.075 1881.48,244.166 1882.04,244.214 1882.6,244.221 1883.16,244.185 1883.71,244.109 1884.27,243.991 1884.83,243.832 1885.39,243.634 1885.95,243.396 \n 1886.51,243.119 1887.07,242.804 1887.63,242.453 1888.19,242.066 1888.74,241.643 1889.3,241.188 1889.86,240.7 1890.42,240.18 1890.98,239.632 1891.54,239.055 \n 1892.1,238.452 1892.66,237.823 1893.21,237.172 1893.77,236.499 1894.33,235.806 1894.89,235.095 1895.45,234.368 1896.01,233.626 1896.57,232.873 1897.13,232.108 \n 1897.69,231.336 1898.24,230.556 1898.8,229.773 1899.36,228.986 1899.92,228.199 1900.48,227.413 1901.04,226.631 1901.6,225.854 1902.16,225.084 1902.72,224.323 \n 1903.27,223.573 1903.83,222.835 1904.39,222.112 1904.95,221.406 1905.51,220.717 1906.07,220.048 1906.63,219.4 1907.19,218.776 1907.75,218.175 1908.3,217.6 \n 1908.86,217.052 1909.42,216.532 1909.98,216.042 1910.54,215.582 1911.1,215.154 1911.66,214.758 1912.22,214.396 1912.78,214.068 1913.33,213.775 1913.89,213.518 \n 1914.45,213.296 1915.01,213.112 1915.57,212.963 1916.13,212.852 1916.69,212.778 1917.25,212.742 1917.81,212.742 1918.36,212.78 1918.92,212.854 1919.48,212.965 \n 1920.04,213.112 1920.6,213.295 1921.16,213.512 1921.72,213.764 1922.28,214.05 1922.84,214.369 1923.39,214.719 1923.95,215.1 1924.51,215.511 1925.07,215.951 \n 1925.63,216.418 1926.19,216.912 1926.75,217.431 1927.31,217.973 1927.86,218.537 1928.42,219.122 1928.98,219.726 1929.54,220.347 1930.1,220.985 1930.66,221.636 \n 1931.22,222.301 1931.78,222.976 1932.34,223.66 1932.89,224.352 1933.45,225.05 1934.01,225.751 1934.57,226.455 1935.13,227.159 1935.69,227.862 1936.25,228.562 \n 1936.81,229.256 1937.37,229.945 1937.92,230.625 1938.48,231.295 1939.04,231.953 1939.6,232.599 1940.16,233.229 1940.72,233.844 1941.28,234.44 1941.84,235.018 \n 1942.4,235.575 1942.95,236.11 1943.51,236.622 1944.07,237.109 1944.63,237.572 1945.19,238.007 1945.75,238.415 1946.31,238.795 1946.87,239.145 1947.43,239.465 \n 1947.98,239.755 1948.54,240.013 1949.1,240.239 1949.66,240.432 1950.22,240.593 1950.78,240.721 1951.34,240.815 1951.9,240.876 1952.46,240.903 1953.01,240.897 \n 1953.57,240.858 1954.13,240.785 1954.69,240.68 1955.25,240.542 1955.81,240.372 1956.37,240.171 1956.93,239.939 1957.48,239.676 1958.04,239.384 1958.6,239.064 \n 1959.16,238.715 1959.72,238.34 1960.28,237.939 1960.84,237.513 1961.4,237.064 1961.96,236.592 1962.51,236.099 1963.07,235.587 1963.63,235.056 1964.19,234.508 \n 1964.75,233.944 1965.31,233.365 1965.87,232.775 1966.43,232.172 1966.99,231.561 1967.54,230.941 1968.1,230.314 1968.66,229.683 1969.22,229.048 1969.78,228.411 \n 1970.34,227.774 1970.9,227.139 1971.46,226.506 1972.02,225.878 1972.57,225.256 1973.13,224.642 1973.69,224.037 1974.25,223.442 1974.81,222.86 1975.37,222.291 \n 1975.93,221.736 1976.49,221.199 1977.05,220.678 1977.6,220.176 1978.16,219.695 1978.72,219.234 1979.28,218.796 1979.84,218.38 1980.4,217.989 1980.96,217.623 \n 1981.52,217.283 1982.08,216.97 1982.63,216.684 1983.19,216.426 1983.75,216.196 1984.31,215.996 1984.87,215.825 1985.43,215.684 1985.99,215.573 1986.55,215.493 \n 1987.1,215.442 1987.66,215.423 1988.22,215.433 1988.78,215.474 1989.34,215.545 1989.9,215.646 1990.46,215.776 1991.02,215.936 1991.58,216.124 1992.13,216.34 \n 1992.69,216.583 1993.25,216.853 1993.81,217.15 1994.37,217.471 1994.93,217.817 1995.49,218.186 1996.05,218.577 1996.61,218.99 1997.16,219.423 1997.72,219.876 \n 1998.28,220.346 1998.84,220.833 1999.4,221.335 1999.96,221.851 2000.52,222.381 2001.08,222.921 2001.64,223.472 2002.19,224.032 2002.75,224.598 2003.31,225.171 \n 2003.87,225.748 2004.43,226.328 2004.99,226.909 2005.55,227.491 2006.11,228.071 2006.67,228.648 2007.22,229.221 2007.78,229.788 2008.34,230.348 2008.9,230.899 \n 2009.46,231.441 2010.02,231.971 2010.58,232.489 2011.14,232.993 2011.7,233.482 2012.25,233.955 2012.81,234.411 2013.37,234.849 2013.93,235.267 2014.49,235.664 \n 2015.05,236.041 2015.61,236.395 2016.17,236.726 2016.73,237.033 2017.28,237.316 2017.84,237.574 2018.4,237.805 2018.96,238.011 2019.52,238.19 2020.08,238.342 \n 2020.64,238.467 2021.2,238.563 2021.75,238.633 2022.31,238.674 2022.87,238.687 2023.43,238.673 2023.99,238.631 2024.55,238.561 2025.11,238.464 2025.67,238.34 \n 2026.23,238.189 2026.78,238.013 2027.34,237.81 2027.9,237.583 2028.46,237.33 2029.02,237.055 2029.58,236.756 2030.14,236.435 2030.7,236.092 2031.26,235.73 \n 2031.81,235.347 2032.37,234.946 2032.93,234.528 2033.49,234.094 2034.05,233.644 2034.61,233.18 2035.17,232.703 2035.73,232.215 2036.29,231.716 2036.84,231.209 \n 2037.4,230.693 2037.96,230.171 2038.52,229.643 2039.08,229.112 2039.64,228.578 2040.2,228.044 2040.76,227.509 2041.32,226.975 2041.87,226.445 2042.43,225.919 \n 2042.99,225.398 2043.55,224.884 2044.11,224.377 2044.67,223.88 2045.23,223.394 2045.79,222.919 2046.35,222.457 2046.9,222.008 2047.46,221.575 2048.02,221.158 \n 2048.58,220.758 2049.14,220.376 2049.7,220.013 2050.26,219.669 2050.82,219.346 2051.37,219.045 2051.93,218.765 2052.49,218.508 2053.05,218.275 2053.61,218.065 \n 2054.17,217.879 2054.73,217.718 2055.29,217.582 2055.85,217.472 2056.4,217.387 2056.96,217.327 2057.52,217.293 2058.08,217.285 2058.64,217.303 2059.2,217.346 \n 2059.76,217.415 2060.32,217.508 2060.88,217.627 2061.43,217.77 2061.99,217.938 2062.55,218.129 2063.11,218.343 2063.67,218.579 2064.23,218.838 2064.79,219.118 \n 2065.35,219.418 2065.91,219.738 2066.46,220.076 2067.02,220.432 2067.58,220.806 2068.14,221.195 2068.7,221.599 2069.26,222.018 2069.82,222.449 2070.38,222.892 \n 2070.94,223.345 2071.49,223.808 2072.05,224.279 2072.61,224.758 2073.17,225.242 2073.73,225.73 2074.29,226.223 2074.85,226.717 2075.41,227.212 2075.97,227.707 \n 2076.52,228.201 2077.08,228.692 2077.64,229.178 2078.2,229.66 2078.76,230.135 2079.32,230.603 2079.88,231.062 2080.44,231.511 2081,231.949 2081.55,232.375 \n 2082.11,232.788 2082.67,233.188 2083.23,233.572 2083.79,233.94 2084.35,234.291 2084.91,234.625 2085.47,234.941 2086.02,235.237 2086.58,235.513 2087.14,235.769 \n 2087.7,236.004 2088.26,236.217 2088.82,236.408 2089.38,236.576 2089.94,236.722 2090.5,236.844 2091.05,236.943 2091.61,237.018 2092.17,237.069 2092.73,237.096 \n 2093.29,237.1 2093.85,237.079 2094.41,237.035 2094.97,236.968 2095.53,236.877 2096.08,236.762 2096.64,236.625 2097.2,236.466 2097.76,236.285 2098.32,236.083 \n 2098.88,235.859 2099.44,235.616 2100,235.352 2100.56,235.07 2101.11,234.77 2101.67,234.452 2102.23,234.118 2102.79,233.769 2103.35,233.404 2103.91,233.026 \n 2104.47,232.635 2105.03,232.232 2105.59,231.818 2106.14,231.395 2106.7,230.962 2107.26,230.523 2107.82,230.077 2108.38,229.625 2108.94,229.17 2109.5,228.711 \n 2110.06,228.251 2110.62,227.79 2111.17,227.329 2111.73,226.87 2112.29,226.413 2112.85,225.961 2113.41,225.513 2113.97,225.072 2114.53,224.637 2115.09,224.211 \n 2115.64,223.794 2116.2,223.388 2116.76,222.993 2117.32,222.61 2117.88,222.24 2118.44,221.884 2119,221.544 2119.56,221.219 2120.12,220.911 2120.67,220.619 \n 2121.23,220.346 2121.79,220.092 2122.35,219.857 2122.91,219.641 2123.47,219.446 2124.03,219.271 2124.59,219.118 2125.15,218.986 2125.7,218.875 2126.26,218.787 \n 2126.82,218.72 2127.38,218.676 2127.94,218.655 2128.5,218.655 2129.06,218.678 2129.62,218.723 2130.18,218.79 2130.73,218.878 2131.29,218.988 2131.85,219.12 \n 2132.41,219.272 2132.97,219.444 2133.53,219.637 2134.09,219.849 2134.65,220.079 2135.21,220.328 2135.76,220.595 2136.32,220.878 2136.88,221.177 2137.44,221.492 \n 2138,221.821 2138.56,222.164 2139.12,222.52 2139.68,222.887 2140.24,223.266 2140.79,223.654 2141.35,224.052 2141.91,224.457 2142.47,224.869 2143.03,225.287 \n 2143.59,225.71 2144.15,226.137 2144.71,226.566 2145.26,226.997 2145.82,227.428 2146.38,227.859 2146.94,228.288 2147.5,228.715 2148.06,229.138 2148.62,229.556 \n 2149.18,229.968 2149.74,230.373 2150.29,230.77 2150.85,231.159 2151.41,231.538 2151.97,231.906 2152.53,232.262 2153.09,232.606 2153.65,232.937 2154.21,233.254 \n 2154.77,233.555 2155.32,233.841 2155.88,234.111 2156.44,234.364 2157,234.6 2157.56,234.817 2158.12,235.016 2158.68,235.195 2159.24,235.356 2159.8,235.496 \n 2160.35,235.617 2160.91,235.717 2161.47,235.796 2162.03,235.855 2162.59,235.893 2163.15,235.909 2163.71,235.905 2164.27,235.881 2164.83,235.835 2165.38,235.769 \n 2165.94,235.682 2166.5,235.576 2167.06,235.449 2167.62,235.303 2168.18,235.138 2168.74,234.955 2169.3,234.753 2169.86,234.533 2170.41,234.297 2170.97,234.044 \n 2171.53,233.776 2172.09,233.493 2172.65,233.195 2173.21,232.884 2173.77,232.56 2174.33,232.224 2174.89,231.877 2175.44,231.52 2176,231.154 2176.56,230.78 \n 2177.12,230.398 2177.68,230.01 2178.24,229.617 2178.8,229.219 2179.36,228.818 2179.91,228.414 2180.47,228.009 2181.03,227.604 2181.59,227.199 2182.15,226.796 \n 2182.71,226.396 2183.27,226 2183.83,225.608 2184.39,225.222 2184.94,224.842 2185.5,224.47 2186.06,224.106 2186.62,223.751 2187.18,223.407 2187.74,223.074 \n 2188.3,222.753 2188.86,222.444 2189.42,222.149 2189.97,221.867 2190.53,221.601 2191.09,221.35 2191.65,221.115 2192.21,220.896 2192.77,220.694 2193.33,220.51 \n 2193.89,220.344 2194.45,220.197 2195,220.068 2195.56,219.958 2196.12,219.867 2196.68,219.795 2197.24,219.743 2197.8,219.711 2198.36,219.699 2198.92,219.706 \n 2199.48,219.732 2200.03,219.778 2200.59,219.844 2201.15,219.929 2201.71,220.032 2202.27,220.154 2202.83,220.295 2203.39,220.453 2203.95,220.629 2204.51,220.822 \n 2205.06,221.031 2205.62,221.256 2206.18,221.497 2206.74,221.752 2207.3,222.022 2207.86,222.304 2208.42,222.6 2208.98,222.907 2209.53,223.225 2210.09,223.553 \n 2210.65,223.891 2211.21,224.237 2211.77,224.592 2212.33,224.952 2212.89,225.319 2213.45,225.691 2214.01,226.066 2214.56,226.445 2215.12,226.826 2215.68,227.207 \n 2216.24,227.589 2216.8,227.971 2217.36,228.35 2217.92,228.727 2218.48,229.101 2219.04,229.469 2219.59,229.833 2220.15,230.19 2220.71,230.539 2221.27,230.881 \n 2221.83,231.214 2222.39,231.537 2222.95,231.849 2223.51,232.151 2224.07,232.44 2224.62,232.716 2225.18,232.98 2225.74,233.229 2226.3,233.463 2226.86,233.683 \n 2227.42,233.886 2227.98,234.074 2228.54,234.245 2229.1,234.398 2229.65,234.535 2230.21,234.654 2230.77,234.755 2231.33,234.837 2231.89,234.902 2232.45,234.948 \n 2233.01,234.975 2233.57,234.984 2234.13,234.974 2234.68,234.946 2235.24,234.899 2235.8,234.834 2236.36,234.751 2236.92,234.65 2237.48,234.532 2238.04,234.396 \n 2238.6,234.244 2239.16,234.075 2239.71,233.89 2240.27,233.69 2240.83,233.474 2241.39,233.245 2241.95,233.001 2242.51,232.744 2243.07,232.475 2243.63,232.194 \n 2244.18,231.902 2244.74,231.599 2245.3,231.287 2245.86,230.967 2246.42,230.638 2246.98,230.302 2247.54,229.96 2248.1,229.612 2248.66,229.26 2249.21,228.904 \n 2249.77,228.546 2250.33,228.185 2250.89,227.824 2251.45,227.463 2252.01,227.102 2252.57,226.743 2253.13,226.387 2253.69,226.035 2254.24,225.687 2254.8,225.344 \n 2255.36,225.007 2255.92,224.677 2256.48,224.355 2257.04,224.042 2257.6,223.738 2258.16,223.444 2258.72,223.16 2259.27,222.889 2259.83,222.629 2260.39,222.382 \n 2260.95,222.148 2261.51,221.929 2262.07,221.724 2262.63,221.533 2263.19,221.358 2263.75,221.199 2264.3,221.056 2264.86,220.93 2265.42,220.82 2265.98,220.728 \n 2266.54,220.652 2267.1,220.594 2267.66,220.553 2268.22,220.531 2268.78,220.525 2269.33,220.537 2269.89,220.567 2270.45,220.614 2271.01,220.679 2271.57,220.76 \n 2272.13,220.859 2272.69,220.974 2273.25,221.105 2273.8,221.252 2274.36,221.415 2274.92,221.592 2275.48,221.785 2276.04,221.991 2276.6,222.211 2277.16,222.444 \n 2277.72,222.69 2278.28,222.947 2278.83,223.215 2279.39,223.494 2279.95,223.782 2280.51,224.079 2281.07,224.385 2281.63,224.698 2282.19,225.018 2282.75,225.343 \n 2283.31,225.674 2283.86,226.008 2284.42,226.346 2284.98,226.687 2285.54,227.029 2286.1,227.372 2286.66,227.714 2287.22,228.056 2287.78,228.396 2288.34,228.733 \n 2288.89,229.067 2289.45,229.397 2290.01,229.721 2290.57,230.04 2291.13,230.352 2291.69,230.656 2292.25,230.952 2292.81,231.239 2293.37,231.517 2293.92,231.784 \n 2294.48,232.04 2295.04,232.284 2295.6,232.517 2296.16,232.736 2296.72,232.943 2297.28,233.135 2297.84,233.313 2298.4,233.477 2298.95,233.625 2299.51,233.759 \n 2300.07,233.876 2300.63,233.977 2301.19,234.063 2301.75,234.132 2302.31,234.184 2302.87,234.22 2303.42,234.239 2303.98,234.241 2304.54,234.227 2305.1,234.196 \n 2305.66,234.148 2306.22,234.084 2306.78,234.004 2307.34,233.908 2307.9,233.796 2308.45,233.669 2309.01,233.526 2309.57,233.369 2310.13,233.198 2310.69,233.013 \n 2311.25,232.814 2311.81,232.603 2312.37,232.38 2312.93,232.144 2313.48,231.898 2314.04,231.641 2314.6,231.375 2315.16,231.099 2315.72,230.815 2316.28,230.523 \n 2316.84,230.224 2317.4,229.919 2317.96,229.609 2318.51,229.294 2319.07,228.975 2319.63,228.653 2320.19,228.329 2320.75,228.004 2321.31,227.678 2321.87,227.352 \n 2322.43,227.027 2322.99,226.704 2323.54,226.383 2324.1,226.066 2324.66,225.753 2325.22,225.446 2325.78,225.144 2326.34,224.848 2326.9,224.56 2327.46,224.279 \n 2328.02,224.008 2328.57,223.745 2329.13,223.493 2329.69,223.251 2330.25,223.02 2330.81,222.801 2331.37,222.594 2331.93,222.4 2332.49,222.219 2333.05,222.051 \n 2333.6,221.898 2334.16,221.759 2334.72,221.635 2335.28,221.525 2335.84,221.431 2336.4,221.353 2336.96,221.29 2337.52,221.243 2338.07,221.211 2338.63,221.196 \n 2339.19,221.196 2339.75,221.213 2340.31,221.245 2340.87,221.293 2341.43,221.356 2341.99,221.435 2342.55,221.53 2343.1,221.639 2343.66,221.762 2344.22,221.9 \n 2344.78,222.052 2345.34,222.218 2345.9,222.396 2346.46,222.587 2347.02,222.791 2347.58,223.006 2348.13,223.232 2348.69,223.468 2349.25,223.714 2349.81,223.97 \n 2350.37,224.234 2350.93,224.506 2351.49,224.785 2352.05,225.071 2352.61,225.363 2353.16,225.659 2353.72,225.96 2354.28,226.265 2354.84,226.572 2355.4,226.881 \n 2355.96,227.192 2356.52,227.503 2357.08,227.814 2357.64,228.123 2358.19,228.431 2358.75,228.736 2359.31,229.038 2359.87,229.335 2360.43,229.628 2360.99,229.915 \n 2361.55,230.196 2362.11,230.47 2362.67,230.736 2363.22,230.993 2363.78,231.242 2364.34,231.481 2364.9,231.71 2365.46,231.929 2366.02,232.136 2366.58,232.331 \n 2367.14,232.515 2367.69,232.685 2368.25,232.843 2368.81,232.987 2369.37,233.117 2369.93,233.233 2370.49,233.335 2371.05,233.422 2371.61,233.495 2372.17,233.552 \n 2372.72,233.595 2373.28,233.622 2373.84,233.634 2374.4,233.631 2374.96,233.613 2375.52,233.58 2376.08,233.532 2376.64,233.468 2377.2,233.391 2377.75,233.298 \n 2378.31,233.192 2378.87,233.071 2379.43,232.937 2379.99,232.79 2380.55,232.63 2381.11,232.458 2381.67,232.273 2382.23,232.077 2382.78,231.87 2383.34,231.652 \n 2383.9,231.425 2384.46,231.188 2385.02,230.942 2385.58,230.689 2386.14,230.428 2386.7,230.16 2387.26,229.886 2387.81,229.606 2388.37,229.322 2388.93,229.034 \n 2389.49,228.743 2390.05,228.449 2390.61,228.153 2391.17,227.856 2391.73,227.559 2392.29,227.262 2392.84,226.967 2393.4,226.673 2393.96,226.382 2394.52,226.094 \n 2395.08,225.81 2395.64,225.532 2396.2,225.258 2396.76,224.991 2397.32,224.73 2397.87,224.477 2398.43,224.232 2398.99,223.996 2399.55,223.769 2400.11,223.552 \n 2400.67,223.345 2401.23,223.148 2401.79,222.964 2402.34,222.79 2402.9,222.629 2403.46,222.481 2404.02,222.345 2404.58,222.223 2405.14,222.114 2405.7,222.019 \n 2406.26,221.938 2406.82,221.871 2407.37,221.818 2407.93,221.78 2408.49,221.756 2409.05,221.747 2409.61,221.753 2410.17,221.772 2410.73,221.807 2411.29,221.855 \n 2411.85,221.918 2412.4,221.995 2412.96,222.085 2413.52,222.19 2414.08,222.307 2414.64,222.437 2415.2,222.58 2415.76,222.736 2416.32,222.903 2416.88,223.081 \n 2417.43,223.271 2417.99,223.471 2418.55,223.68 2419.11,223.9 2419.67,224.128 2420.23,224.364 2420.79,224.608 2421.35,224.859 2421.91,225.116 2422.46,225.38 \n 2423.02,225.648 2423.58,225.921 2424.14,226.197 2424.7,226.477 2425.26,226.758 2425.82,227.042 2426.38,227.326 2426.94,227.611 2427.49,227.895 2428.05,228.177 \n 2428.61,228.458 2429.17,228.736 2429.73,229.011 2430.29,229.282 2430.85,229.548 2431.41,229.809 2431.96,230.064 2432.52,230.313 2433.08,230.554 2433.64,230.787 \n 2434.2,231.012 2434.76,231.228 2435.32,231.434 2435.88,231.631 2436.44,231.817 2436.99,231.992 2437.55,232.156 2438.11,232.309 2438.67,232.449 2439.23,232.577 \n 2439.79,232.692 2440.35,232.794 2440.91,232.883 2441.47,232.958 2442.02,233.02 2442.58,233.068 2443.14,233.102 2443.7,233.123 2444.26,233.129 2444.82,233.122 \n 2445.38,233.1 2445.94,233.065 2446.5,233.016 2447.05,232.954 2447.61,232.878 2448.17,232.789 2448.73,232.687 2449.29,232.572 2449.85,232.445 2450.41,232.306 \n 2450.97,232.155 2451.53,231.993 2452.08,231.821 2452.64,231.637 2453.2,231.444 2453.76,231.241 2454.32,231.029 2454.88,230.809 2455.44,230.582 2456,230.346 \n 2456.56,230.105 2457.11,229.857 2457.67,229.603 2458.23,229.345 2458.79,229.083 2459.35,228.818 2459.91,228.549 2460.47,228.279 2461.03,228.007 2461.59,227.734 \n 2462.14,227.461 2462.7,227.189 2463.26,226.918 2463.82,226.649 2464.38,226.382 2464.94,226.119 2465.5,225.86 2466.06,225.605 2466.61,225.356 2467.17,225.112 \n 2467.73,224.875 2468.29,224.645 2468.85,224.423 2469.41,224.208 2469.97,224.002 2470.53,223.806 2471.09,223.619 2471.64,223.442 2472.2,223.276 2472.76,223.12 \n 2473.32,222.976 2473.88,222.843 2474.44,222.723 2475,222.614 2475.56,222.519 2476.12,222.435 2476.67,222.365 2477.23,222.308 2477.79,222.264 2478.35,222.233 \n 2478.91,222.216 2479.47,222.212 2480.03,222.221 2480.59,222.244 2481.15,222.28 2481.7,222.329 2482.26,222.391 2482.82,222.466 2483.38,222.554 2483.94,222.654 \n 2484.5,222.766 2485.06,222.89 2485.62,223.026 2486.18,223.172 2486.73,223.33 2487.29,223.497 2487.85,223.675 2488.41,223.862 2488.97,224.058 2489.53,224.263 \n 2490.09,224.476 2490.65,224.696 2491.21,224.923 2491.76,225.156 2492.32,225.395 2492.88,225.639 2493.44,225.888 2494,226.14 2494.56,226.396 2495.12,226.654 \n 2495.68,226.914 2496.23,227.176 2496.79,227.438 2497.35,227.7 2497.91,227.962 2498.47,228.222 2499.03,228.48 2499.59,228.735 2500.15,228.988 2500.71,229.236 \n 2501.26,229.48 2501.82,229.719 2502.38,229.952 2502.94,230.178 2503.5,230.398 2504.06,230.611 2504.62,230.816 2505.18,231.012 2505.74,231.199 2506.29,231.378 \n 2506.85,231.546 2507.41,231.704 2507.97,231.852 2508.53,231.989 2509.09,232.114 2509.65,232.228 2510.21,232.33 2510.77,232.42 2511.32,232.498 2511.88,232.564 \n 2512.44,232.616 2513,232.657 2513.56,232.684 2514.12,232.698 2514.68,232.7 2515.24,232.689 2515.8,232.665 2516.35,232.628 2516.91,232.579 2517.47,232.517 \n 2518.03,232.443 2518.59,232.357 2519.15,232.258 2519.71,232.149 2520.27,232.028 2520.83,231.895 2521.38,231.753 2521.94,231.6 2522.5,231.437 2523.06,231.264 \n 2523.62,231.082 2524.18,230.892 2524.74,230.694 2525.3,230.488 2525.85,230.275 2526.41,230.056 2526.97,229.831 2527.53,229.6 2528.09,229.364 2528.65,229.124 \n 2529.21,228.881 2529.77,228.634 2530.33,228.386 2530.88,228.135 2531.44,227.883 2532,227.631 2532.56,227.379 2533.12,227.127 2533.68,226.877 2534.24,226.629 \n 2534.8,226.384 2535.36,226.142 2535.91,225.903 2536.47,225.669 2537.03,225.441 2537.59,225.217 2538.15,225 2538.71,224.789 2539.27,224.586 2539.83,224.39 \n 2540.39,224.203 2540.94,224.024 2541.5,223.854 2542.06,223.693 2542.62,223.543 2543.18,223.402 2543.74,223.272 2544.3,223.153 2544.86,223.046 2545.42,222.949 \n 2545.97,222.864 2546.53,222.791 2547.09,222.731 2547.65,222.682 2548.21,222.645 2548.77,222.621 2549.33,222.609 2549.89,222.609 2550.45,222.622 2551,222.648 \n 2551.56,222.685 2552.12,222.734 2552.68,222.796 2553.24,222.869 2553.8,222.954 2554.36,223.051 2554.92,223.158 2555.48,223.277 2556.03,223.406 2556.59,223.545 \n 2557.15,223.694 2557.71,223.853 2558.27,224.02 2558.83,224.197 2559.39,224.381 2559.95,224.574 2560.5,224.773 2561.06,224.979 2561.62,225.192 2562.18,225.41 \n 2562.74,225.633 2563.3,225.861 2563.86,226.092 2564.42,226.327 2564.98,226.565 2565.53,226.805 2566.09,227.047 2566.65,227.29 2567.21,227.533 2567.77,227.776 \n 2568.33,228.018 2568.89,228.259 2569.45,228.497 2570.01,228.733 2570.56,228.966 2571.12,229.195 2571.68,229.42 2572.24,229.64 2572.8,229.854 2573.36,230.062 \n 2573.92,230.264 2574.48,230.459 2575.04,230.646 2575.59,230.826 2576.15,230.997 2576.71,231.159 2577.27,231.312 2577.83,231.456 2578.39,231.589 2578.95,231.713 \n 2579.51,231.826 2580.07,231.928 2580.62,232.019 2581.18,232.099 2581.74,232.167 2582.3,232.224 2582.86,232.269 2583.42,232.303 2583.98,232.324 2584.54,232.333 \n 2585.1,232.331 2585.65,232.316 2586.21,232.29 2586.77,232.252 2587.33,232.202 2587.89,232.141 2588.45,232.068 2589.01,231.985 2589.57,231.89 2590.12,231.784 \n 2590.68,231.668 2591.24,231.542 2591.8,231.406 2592.36,231.261 2592.92,231.106 2593.48,230.943 2594.04,230.772 2594.6,230.592 2595.15,230.406 2595.71,230.212 \n 2596.27,230.012 2596.83,229.806 2597.39,229.595 2597.95,229.379 2598.51,229.159 2599.07,228.935 2599.63,228.707 2600.18,228.477 2600.74,228.245 2601.3,228.012 \n 2601.86,227.778 2602.42,227.543 2602.98,227.309 2603.54,227.075 2604.1,226.843 2604.66,226.613 2605.21,226.386 2605.77,226.162 2606.33,225.942 2606.89,225.726 \n 2607.45,225.514 2608.01,225.308 2608.57,225.108 2609.13,224.915 2609.69,224.728 2610.24,224.548 2610.8,224.376 2611.36,224.213 2611.92,224.057 2612.48,223.911 \n 2613.04,223.774 2613.6,223.647 2614.16,223.529 2614.72,223.422 2615.27,223.325 2615.83,223.239 2616.39,223.164 2616.95,223.1 2617.51,223.047 2618.07,223.005 \n 2618.63,222.975 2619.19,222.956 2619.75,222.949 2620.3,222.953 2620.86,222.969 2621.42,222.997 2621.98,223.035 2622.54,223.085 2623.1,223.146 2623.66,223.218 \n 2624.22,223.301 2624.77,223.394 2625.33,223.498 2625.89,223.612 2626.45,223.735 2627.01,223.868 2627.57,224.01 2628.13,224.16 2628.69,224.319 2629.25,224.486 \n 2629.8,224.661 2630.36,224.842 2630.92,225.03 2631.48,225.224 2632.04,225.424 2632.6,225.629 2633.16,225.838 2633.72,226.052 2634.28,226.269 2634.83,226.489 \n 2635.39,226.711 2635.95,226.936 2636.51,227.161 2637.07,227.388 2637.63,227.615 2638.19,227.841 2638.75,228.066 2639.31,228.29 2639.86,228.512 2640.42,228.731 \n 2640.98,228.947 2641.54,229.159 2642.1,229.367 2642.66,229.57 2643.22,229.768 2643.78,229.961 2644.34,230.147 2644.89,230.326 2645.45,230.498 2646.01,230.663 \n 2646.57,230.82 2647.13,230.969 2647.69,231.109 2648.25,231.239 2648.81,231.361 2649.37,231.473 2649.92,231.575 2650.48,231.667 2651.04,231.748 2651.6,231.819 \n 2652.16,231.879 2652.72,231.929 2653.28,231.967 2653.84,231.994 2654.39,232.011 2654.95,232.016 2655.51,232.01 2656.07,231.992 2656.63,231.964 2657.19,231.925 \n 2657.75,231.875 2658.31,231.814 2658.87,231.743 2659.42,231.661 2659.98,231.569 2660.54,231.467 2661.1,231.356 2661.66,231.235 2662.22,231.105 2662.78,230.966 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| "metadata": {}, | |
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| "source": "#### Dirichlet積分が絶対収束しないことの証明\n\n**問題:** $\\ds\\int_0^\\infty \\left|\\frac{\\sin x}{x}\\right|\\,dx=\\infty$ を示せ.\n\n**解答例:** $|\\sin(x+\\pi)|=|\\sin x|$ と $\\int_0^\\pi \\sin x\\,dx = 2$ より,\n\n$$\n\\begin{aligned}\n\\int_0^\\infty \\left|\\frac{\\sin x}{x}\\right|\\,dx &=\n\\sum_{n=1}^\\infty \\int_{(n-1)\\pi}^{n\\pi} \\frac{|\\sin x|}{x}\\,dx \\geqq\n\\sum_{n=1}^\\infty \\int_{(n-1)\\pi}^{n\\pi} \\frac{|\\sin x|}{n\\pi}\\,dx \n\\\\ &=\n\\sum_{n=1}^\\infty \\frac{1}{n\\pi}\\int_0^\\pi \\sin x\\,dx =\n\\frac{2}{\\pi}\\sum_{n=1}^\\infty \\frac{1}{n} = \\infty.\n\\qquad \\QED\n\\end{aligned}\n$$\n\n**注意:** 上の解答例の計算の仕方より,\n\n$$\n\\int_0^{N\\pi}\\left|\\frac{\\sin x}{x}\\right|\\,dx \\geqq\n\\frac{2}{\\pi}\\sum_{n=1}^N\\frac{1}{n} \\sim \\frac{2}{\\pi}\\log N\n$$\n\nとなることがわかる." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Dirichlet積分が絶対収束しない様子\n\n**問題:** $\\ds G(R) = \\int_0^R \\left|\\frac{\\sin x}{x}\\right|\\,dx$ をプロットせよ. $\\QED$\n\n次のセルを見よ." | |
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| "metadata": { | |
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| "cell_type": "code", | |
| "source": "g(x) = abs(iszero(x) ? one(x) : sin(x)/x)\nG(R) = quadgk(g, 0, R)[1]\n@show G(π)\n\nN = 1:0.1:10\nR = N.*π\nP1 = plot(size=(600, 300), legend=:topleft, xscale=:log)\nplot!(R, G.(R), label=\"G(R)\", lw=2)\nplot!(R, @.(2/π*log(R/π)+G(π)), label=\"(2/pi) log(R/pi) + G(pi)\", ls=:dash, lw=2)\n\nN = 1:100\nR = N.*π\nP2 = plot(size=(600, 300), legend=:topleft, xscale=:log)\nplot!(R, G.(R), label=\"G(R)\", lw=2)\nplot!(R, @.(2/π*log(R/π)+G(π)), label=\"(2/pi) log(R/pi) + G(pi)\", ls=:dash, lw=2)\n\nplot(P1, P2, size=(800, 300))", | |
| "execution_count": 12, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "G(π) = 1.8519370519824665\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 12, | |
| "data": { | |
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| "source": "#### Fresnel積分\n\n**Fresnel積分:** 次の不定積分をFresnel積分(フレネル積分)と呼ぶ:\n\n$$\nC(t) = \\int_0^t \\cos(s^2)\\,ds, \\quad S(t) = \\int_0^t \\sin(s^2)\\,ds.\n$$\n\n上で紹介したように, 条件収束する広義積分の例として, \n\n$$\n\\lim_{t\\to^\\infty}C(t)=\n\\lim_{t\\to^\\infty}S(t)=\n\\frac{1}{2}\\sqrt{\\frac{\\pi}{2}}\n$$\n\nは有名である. $C'(t)^2+S'(t)^2=1$ なので点 $(C(t),S(t))$ の運動の速さは常に $1$ である. だから時刻 $0$ から $t$ までの軌跡の長さは $t$ に等しい. だから軌跡の長さは $t\\to\\infty$ で無限大に発散する. 無限に長い軌跡の最終到着地点が $a=(1/2)\\sqrt{\\pi/2}$ に対する点 $(a,a)$ になる." | |
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| "source": "#### Fresnel積分の収束の様子\n\nFresnel積分 $C(t),S(t)$ のそれぞれをプロットすると以下のようになる. それらが $t\\to\\infty$ で $a=(1/2)\\sqrt{\\pi/2}$ に収束する様子も確認できる." | |
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| "source": "C(t) = quadgk(s->cos(s^2), 0, t)[1]\nS(t) = quadgk(s->sin(s^2), 0, t)[1]\na = sqrt(π/2)/2\n\nPP = []\n\ntmin, tmax = 0, 20\nt = tmin:(tmax-tmin)/1000:tmax\nPC = plot(t, C.(t), label=\"C(t)\"); hline!([a], label=\"a\")\nPS = plot(t, C.(t), label=\"S(t)\"); hline!([a], label=\"a\")\npush!(PP, PC, PS)\n\nplot(PP..., size=(700, 250), legend=:bottomright)", | |
| "execution_count": 13, | |
| "outputs": [ | |
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| "execution_count": 13, | |
| "data": { | |
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411.596,433.163 412.694,418.831 413.792,403.484 414.89,387.417 415.988,370.942 417.087,354.386 418.185,338.083 \n 419.283,322.367 420.381,307.566 421.479,293.993 422.578,281.942 423.676,271.678 424.774,263.432 425.872,257.395 426.97,253.711 428.069,252.476 429.167,253.73 \n 430.265,257.458 431.363,263.586 432.461,271.985 433.56,282.469 434.658,294.801 435.756,308.693 436.854,323.819 437.952,339.812 439.051,356.284 440.149,372.823 \n 441.247,389.015 442.345,404.444 443.444,418.709 444.542,431.434 445.64,442.278 446.738,450.944 447.836,457.189 448.935,460.832 450.033,461.763 451.131,459.942 \n 452.229,455.407 453.327,448.273 454.426,438.729 455.524,427.038 456.622,413.528 457.72,398.583 458.818,382.639 459.917,366.163 461.015,349.647 462.113,333.592 \n 463.211,318.489 464.309,304.809 465.408,292.984 466.506,283.392 467.604,276.348 468.702,272.088 469.8,270.761 470.899,272.423 471.997,277.034 473.095,284.452 \n 474.193,294.443 475.291,306.681 476.39,320.759 477.488,336.204 478.586,352.486 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547.773,407.89 548.871,418.869 549.969,426.616 \n 551.067,430.679 552.166,430.814 553.264,427.001 554.362,419.448 555.46,408.584 556.558,395.037 557.657,379.599 558.755,363.183 559.853,346.77 560.951,331.35 \n 562.049,317.861 563.148,307.134 564.246,299.838 565.344,296.433 566.442,297.144 567.54,301.94 568.639,310.53 569.737,322.381 570.835,336.746 571.933,352.71 \n 573.032,369.245 574.13,385.276 575.228,399.753 576.326,411.717 577.424,420.367 578.523,425.117 579.621,425.635 580.719,421.875 581.817,414.079 582.915,402.769 \n 584.014,388.71 585.112,372.869 586.21,356.342 587.308,340.286 588.406,325.831 589.505,314.007 590.603,305.665 591.701,301.413 592.799,301.569 593.897,306.135 \n 594.996,314.789 596.094,326.909 597.192,341.608 598.29,357.803 599.388,374.287 600.487,389.823 601.585,403.234 602.683,413.494 603.781,419.81 604.879,421.686 \n 605.978,418.964 607.076,411.844 608.174,400.869 609.272,386.894 610.37,371.015 611.469,354.49 612.567,338.639 613.665,324.741 614.763,313.923 615.861,307.074 \n 616.96,304.766 618.058,307.2 619.156,314.188 620.254,325.16 621.352,339.21 622.451,355.164 623.549,371.678 624.647,387.351 625.745,400.84 626.843,410.982 \n 627.942,416.89 629.04,418.041 630.138,414.322 631.236,406.047 632.334,393.935 633.433,379.051 634.531,362.714 635.629,346.388 636.727,331.542 637.826,319.528 \n 638.924,311.445 640.022,308.045 641.12,309.653 642.218,316.131 643.317,326.89 644.415,340.934 645.513,356.951 646.611,373.433 647.709,388.816 648.808,401.629 \n 649.906,410.636 651.004,414.96 652.102,414.169 653.2,408.33 654.299,398 655.397,384.186 656.495,368.245 657.593,351.758 658.691,336.368 659.79,323.625 \n 660.888,314.82 661.986,310.855 663.084,312.146 664.182,318.574 665.281,329.487 666.379,343.77 667.477,359.944 668.575,376.326 669.673,391.196 670.772,402.984 \n 671.87,410.431 672.968,412.736 674.066,409.64 675.164,401.464 676.263,389.078 677.361,373.818 678.459,357.343 679.557,341.454 680.655,327.903 681.754,318.195 \n 682.852,313.418 683.95,314.116 685.048,320.223 686.146,331.063 687.245,345.418 688.343,361.663 689.441,377.943 690.539,392.389 691.637,403.329 692.736,409.488 \n 693.834,410.136 694.932,405.186 696.03,395.208 697.128,381.369 698.227,365.303 699.325,348.921 700.423,334.182 701.521,322.865 702.62,316.342 703.718,315.418 \n 704.816,320.215 705.914,330.16 707.012,344.039 708.111,360.145 709.209,376.484 710.307,391.018 711.405,401.924 712.503,407.821 713.602,407.955 714.7,402.295 \n 715.798,391.551 716.896,377.089 717.994,360.761 719.093,344.672 720.191,330.911 721.289,321.276 722.387,317.036 723.485,318.759 724.584,326.232 725.682,338.473 \n 726.78,353.863 727.878,370.35 728.976,385.721 730.075,397.901 731.173,405.232 732.271,406.708 733.369,402.116 734.467,392.072 735.566,377.951 736.664,361.699 \n 737.762,345.569 738.86,331.814 739.958,322.366 741.057,318.563 742.155,320.953 743.253,329.21 744.351,342.168 745.449,357.982 746.548,374.386 747.646,389.012 \n 748.744,399.74 749.842,404.999 750.94,404.012 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820.127,382.995 821.225,367.556 822.324,351.166 823.422,336.909 824.52,327.481 \n 825.618,324.679 826.716,329.047 827.815,339.763 828.913,354.782 830.011,371.223 831.109,385.914 832.208,396.007 833.306,399.531 834.404,395.788 835.502,385.5 \n 836.6,370.677 837.699,354.235 838.797,339.425 839.895,329.19 840.993,325.577 842.091,329.319 843.19,339.677 844.288,354.575 845.386,371.011 846.484,385.655 \n 847.582,395.523 848.681,398.595 849.779,394.229 850.877,383.312 851.975,368.086 853.073,351.695 854.172,337.542 855.27,328.58 856.368,326.692 857.466,332.285 \n 858.564,344.191 859.663,359.907 860.761,376.106 861.859,389.343 862.957,396.789 864.055,396.841 865.154,389.476 866.252,376.27 867.35,360.072 868.448,344.392 \n 869.546,332.645 870.645,327.404 871.743,329.827 872.841,339.393 873.939,354.002 875.037,370.427 876.136,385.022 877.234,394.531 878.332,396.822 879.43,391.369 \n 880.528,379.388 881.627,363.575 882.725,347.507 883.823,334.835 884.921,328.454 886.019,329.833 887.118,338.667 888.216,352.932 889.314,369.341 890.412,384.095 \n 891.51,393.76 892.609,396.076 893.707,390.489 894.805,378.299 895.903,362.369 897.001,346.456 898.1,334.332 899.198,328.884 900.296,331.423 901.394,341.351 \n 902.493,356.296 903.591,372.667 904.689,386.51 905.787,394.468 906.885,394.597 907.984,386.856 909.082,373.127 910.18,356.77 911.278,341.809 912.376,331.939 \n 913.475,329.612 914.573,335.418 915.671,347.919 916.769,364.001 917.867,379.634 918.966,390.885 920.064,394.91 921.162,390.679 922.26,379.255 923.358,363.538 \n 924.457,347.538 925.555,335.354 926.653,330.122 927.751,333.202 928.849,343.807 929.948,359.198 931.046,375.376 932.144,388.119 933.242,394.089 934.34,391.708 \n 935.439,381.593 936.537,366.404 937.635,350.16 938.733,337.175 939.831,330.914 940.93,333.061 942.028,343.05 943.126,358.206 944.224,374.447 945.322,387.381 \n 946.421,393.494 947.519,391.114 948.617,380.88 949.715,365.582 950.813,349.414 951.912,336.828 953.01,331.305 954.108,334.382 955.206,345.217 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1024.39,359.105 1025.49,344.106 1026.59,335.196 1027.69,335.356 1028.79,344.544 1029.88,359.686 1030.98,375.692 1032.08,387.162 1033.18,390.213 \n 1034.28,383.798 1035.38,370.089 1036.47,353.753 1037.57,340.372 1038.67,334.535 1039.77,338.257 1040.87,350.266 1041.96,366.422 1043.06,381.134 1044.16,389.292 \n 1045.26,388.048 1046.36,377.825 1047.46,362.193 1048.55,346.63 1049.65,336.61 1050.75,335.674 1051.85,344.163 1052.95,359.077 1054.04,375.122 1055.14,386.583 \n 1056.24,389.36 1057.34,382.45 1058.44,368.323 1059.54,352.056 1060.63,339.52 1061.73,335.252 1062.83,340.81 1063.93,354.184 1065.03,370.505 1066.12,383.812 \n 1067.22,389.225 1068.32,384.748 1069.42,372.02 1070.52,355.73 1071.62,341.904 1072.71,335.675 1073.81,339.367 1074.91,351.615 1076.01,367.848 1077.11,381.986 \n 1078.21,388.717 1079.3,385.497 1080.4,373.532 1081.5,357.341 1082.6,343.065 1083.7,336.134 1084.79,339.2 1085.89,351.1 1086.99,367.284 1088.09,381.539 \n 1089.19,388.373 1090.29,385.142 1091.38,373.084 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y=\"695.165\">a</text>\n</g>\n</svg>\n" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "次のセルで $(C(t),S(t))$ の軌跡をプロットしてみよう." | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "C(t) = quadgk(s->cos(s^2), 0, t)[1]\nS(t) = quadgk(s->sin(s^2), 0, t)[1]\na = sqrt(π/2)/2\n\nP = []\nfor (lw, tmax) in [(1,7), (0.6,10), (0.4,20), (0.25,40)]\n t = 0:0.01:tmax\n p = plot(xlims=(0,1), ylims=(0,1), legend=false)\n plot!(title=\"Euler spiral (t = 0 to $tmax)\", titlefontsize=10)\n plot!(C.(t), S.(t), lw=lw)\n scatter!([a],[a])\n push!(P, p)\nend\n#pngplot(P..., size=(600, 600))\nplot(P..., size=(600, 600))", | |
| "execution_count": 14, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 14, | |
| "data": { | |
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2154.79,450.601 2152.98,440.538 2150.61,430.597 2147.7,420.81 2144.25,411.211 2140.27,401.832 2135.77,392.703 \n 2130.77,383.858 2125.27,375.325 2119.31,367.136 2112.89,359.319 2106.03,351.903 2098.77,344.914 2091.11,338.377 2083.09,332.318 2074.74,326.76 2066.08,321.723 \n 2057.15,317.228 2047.97,313.292 2038.57,309.932 2028.99,307.161 2019.27,304.992 2009.44,303.435 1999.54,302.497 1989.59,302.185 1979.65,302.5 1969.75,303.445 \n 1959.92,305.018 1950.2,307.214 1940.64,310.027 1931.26,313.448 1922.12,317.466 1913.23,322.067 1904.65,327.234 1896.39,332.949 1888.51,339.19 1881.03,345.934 \n 1873.98,353.155 1867.39,360.826 1861.3,368.915 1855.73,377.39 1850.7,386.219 1846.23,395.364 1842.35,404.788 1839.08,414.452 1836.43,424.316 1834.41,434.337 \n 1833.03,444.473 1832.31,454.68 1832.25,464.914 1832.85,475.129 1834.1,485.282 1836.02,495.325 1838.58,505.214 1841.78,514.903 1845.61,524.349 1850.05,533.507 \n 1855.08,542.335 1860.68,550.79 1866.83,558.833 1873.5,566.424 1880.66,573.527 1888.28,580.107 1896.32,586.131 1904.75,591.568 1913.52,596.391 1922.6,600.575 \n 1931.94,604.097 1941.49,606.938 1951.22,609.083 1961.07,610.518 1970.99,611.236 1980.94,611.229 1990.86,610.496 2000.7,609.039 2010.42,606.863 2019.96,603.977 \n 2029.28,600.395 2038.32,596.132 2047.04,591.209 2055.39,585.649 2063.33,579.482 2070.81,572.736 2077.79,565.448 2084.24,557.653 2090.11,549.393 2095.38,540.712 \n 2100.01,531.654 2103.97,522.269 2107.25,512.608 2109.82,502.722 2111.67,492.667 2112.78,482.497 2113.15,472.271 2112.77,462.045 2111.65,451.877 2109.78,441.826 \n 2107.18,431.949 2103.85,422.305 2099.82,412.948 2095.11,403.936 2089.74,395.32 2083.75,387.154 2077.16,379.487 2070.02,372.365 2062.36,365.833 2054.24,359.931 \n 2045.69,354.696 2036.77,350.163 2027.54,346.361 2018.04,343.314 2008.35,341.043 1998.5,339.565 1988.58,338.891 1978.64,339.027 1968.73,339.974 1958.93,341.728 \n 1949.3,344.28 1939.9,347.616 1930.79,351.717 1922.02,356.557 1913.67,362.107 1905.77,368.333 1898.4,375.196 1891.58,382.652 1885.38,390.653 1879.84,399.147 \n 1874.98,408.079 1870.86,417.391 1867.49,427.019 1864.9,436.9 1863.12,446.966 1862.15,457.151 1862,467.383 1862.68,477.592 1864.19,487.707 1866.51,497.657 \n 1869.63,507.373 1873.53,516.785 1878.19,525.826 1883.57,534.431 1889.64,542.537 1896.36,550.085 1903.67,557.019 1911.53,563.287 1919.89,568.843 1928.67,573.644 \n 1937.82,577.652 1947.27,580.837 1956.95,583.174 1966.8,584.641 1976.73,585.227 1986.67,584.926 1996.54,583.737 2006.28,581.667 2015.81,578.73 2025.05,574.947 \n 2033.93,570.345 2042.39,564.958 2050.35,558.825 2057.76,551.993 2064.54,544.514 2070.66,536.444 2076.05,527.848 2080.68,518.791 2084.51,509.346 2087.49,499.586 \n 2089.62,489.59 2090.86,479.437 2091.21,469.211 2090.66,458.994 2089.22,448.87 2086.88,438.923 2083.68,429.235 2079.64,419.888 2074.78,410.959 2069.15,402.524 \n 2062.79,394.657 2055.76,387.423 2048.11,380.887 2039.9,375.105 2031.21,370.129 2022.11,366.002 2012.68,362.763 2002.99,360.441 1993.14,359.059 1983.2,358.63 \n 1973.27,359.16 1963.43,360.648 1953.77,363.081 1944.38,366.44 1935.33,370.697 1926.72,375.816 1918.62,381.751 1911.11,388.453 1904.25,395.86 1898.1,403.907 \n 1892.74,412.522 1888.2,421.627 1884.53,431.137 1881.77,440.967 1879.95,451.025 1879.07,461.217 1879.16,471.449 1880.22,481.623 1882.23,491.644 1885.18,501.415 \n 1889.04,510.842 1893.79,519.836 1899.36,528.308 1905.72,536.175 1912.8,543.36 1920.54,549.791 1928.85,555.405 1937.66,560.143 1946.89,563.958 1956.44,566.81 \n 1966.22,568.669 1976.13,569.515 1986.07,569.336 1995.95,568.132 2005.66,565.915 2015.1,562.703 2024.18,558.528 2032.8,553.431 2040.88,547.462 2048.32,540.681 \n 2055.06,533.157 2061.02,524.965 2066.14,516.191 2070.35,506.925 2073.62,497.263 2075.92,487.307 2077.2,477.162 2077.46,466.934 2076.7,456.733 2074.91,446.668 \n 2072.12,436.848 2068.35,427.379 2063.65,418.364 2058.06,409.903 2051.64,402.089 2044.46,395.009 2036.6,388.742 2028.14,383.358 2019.18,378.92 2009.82,375.478 \n 2000.15,373.073 1990.3,371.733 1980.35,371.477 1970.44,372.308 1960.68,374.219 1951.16,377.19 1942.01,381.19 1933.32,386.175 1925.21,392.089 1917.76,398.865 \n 1911.06,406.427 1905.19,414.687 1900.23,423.551 1896.22,432.915 1893.23,442.671 1891.28,452.704 1890.41,462.895 1890.62,473.124 1891.91,483.268 1894.28,493.205 \n 1897.68,502.816 1902.1,511.984 1907.46,520.596 1913.72,528.548 1920.79,535.741 1928.59,542.085 1937.03,547.502 1946,551.921 1955.39,555.287 1965.08,557.557 \n 1974.96,558.699 1984.91,558.698 1994.79,557.552 2004.48,555.272 2013.86,551.886 2022.82,547.435 2031.22,541.974 2038.98,535.571 2045.98,528.306 2052.13,520.271 \n 2057.36,511.569 2061.6,502.312 2064.77,492.618 2066.86,482.615 2067.82,472.432 2067.63,462.204 2066.31,452.064 2063.87,442.148 2060.33,432.588 2055.74,423.511 \n 2050.16,415.041 2043.67,407.292 2036.35,400.369 2028.3,394.367 2019.62,389.371 2010.44,385.448 2000.88,382.656 1991.06,381.033 1981.13,380.605 1971.21,381.379 \n 1961.45,383.346 1951.99,386.481 1942.95,390.743 1934.46,396.074 1926.65,402.4 1919.62,409.635 1913.48,417.677 1908.3,426.413 1904.18,435.722 1901.16,445.47 \n 1899.3,455.519 1898.63,465.725 1899.14,475.941 1900.85,486.019 1903.72,495.813 1907.71,505.18 1912.78,513.982 1918.84,522.089 1925.82,529.381 1933.6,535.748 \n 1942.08,541.096 1951.12,545.342 1960.6,548.423 1970.38,550.289 1980.3,550.911 1990.22,550.278 2000,548.397 2009.47,545.295 2018.5,541.017 2026.95,535.625 \n 2034.69,529.202 2041.59,521.843 2047.56,513.66 2052.49,504.779 2056.32,495.336 2058.97,485.477 2060.4,475.355 2060.6,465.128 2059.55,454.956 2057.27,444.999 \n 2053.8,435.415 2049.18,426.357 2043.49,417.969 2036.81,410.385 2029.26,403.73 2020.96,398.109 2012.03,393.617 2002.61,390.327 1992.87,388.294 1982.95,387.553 \n 1973.03,388.119 1963.25,389.984 1953.79,393.119 1944.79,397.475 1936.42,402.983 1928.79,409.551 1922.06,417.074 1916.32,425.427 1911.68,434.472 1908.21,444.059 \n 1905.98,454.026 1905.02,464.207 1905.36,474.43 1906.98,484.522 1909.87,494.309 1913.97,503.626 1919.22,512.311 1925.53,520.214 1932.8,527.199 1940.89,533.143 \n 1949.67,537.941 1958.98,541.508 1968.68,543.781 1978.58,544.717 1988.51,544.299 1998.3,542.53 2007.78,539.442 2016.77,535.086 2025.12,529.538 2032.68,522.894 \n 2039.31,515.273 2044.89,506.808 2049.32,497.652 2052.52,487.967 2054.43,477.928 2055,467.717 2054.24,457.519 2052.15,447.521 2048.76,437.904 2044.14,428.847 \n 2038.38,420.516 2031.56,413.067 2023.83,406.638 2015.32,401.351 2006.19,397.305 1996.61,394.579 1986.76,393.224 1976.82,393.268 1966.98,394.713 1957.42,397.533 \n 1948.33,401.677 1939.88,407.068 1932.24,413.604 1925.54,421.164 1919.93,429.603 1915.5,438.76 1912.35,448.46 1910.53,458.516 1910.1,468.733 1911.04,478.914 \n 1913.36,488.86 1917.01,498.376 1921.91,507.274 1927.97,515.379 1935.08,522.53 1943.09,528.584 1951.85,533.419 1961.18,536.938 1970.9,539.069 1980.82,539.766 \n 1990.73,539.014 2000.44,536.827 2009.75,533.246 2018.48,528.341 2026.43,522.211 2033.46,514.978 2039.41,506.789 2044.17,497.809 2047.63,488.221 2049.73,478.223 \n 2050.4,468.02 2049.65,457.822 2047.48,447.842 2043.94,438.286 2039.1,429.355 2033.05,421.237 2025.93,414.102 2017.88,408.103 2009.07,403.368 1999.68,399.997 \n 1989.92,398.065 1979.99,397.615 1970.11,398.658 1960.47,401.174 1951.3,405.111 1942.78,410.386 1935.11,416.887 1928.45,424.476 1922.94,432.99 1918.71,442.244 \n 1915.85,452.038 1914.43,462.159 1914.47,472.386 1915.98,482.493 1918.93,492.259 1923.26,501.467 1928.86,509.912 1935.63,517.406 1943.4,523.78 1952.01,528.892 \n 1961.26,532.625 1970.95,534.893 1980.86,535.643 1990.77,534.857 2000.46,532.55 2009.69,528.772 2018.27,523.608 2026,517.173 2032.69,509.614 2038.2,501.101 \n 2042.39,491.829 2045.17,482.011 2046.47,471.872 2046.26,461.649 2044.54,451.577 2041.35,441.891 2036.76,432.819 2030.89,424.573 2023.85,417.347 2015.83,411.314 \n 2007,406.617 1997.58,403.369 1987.78,401.649 1977.84,401.5 1968,402.926 1958.49,405.897 1949.54,410.343 1941.37,416.158 1934.16,423.206 1928.11,431.316 \n 1923.36,440.295 1920.01,449.925 1918.17,459.972 1917.86,470.193 1919.12,480.337 1921.89,490.156 1926.12,499.408 1931.71,507.864 1938.52,515.313 1946.38,521.572 \n 1955.1,526.481 1964.46,529.918 1974.23,531.795 1984.16,532.064 1994.01,530.714 2003.53,527.778 2012.48,523.329 2020.63,517.477 2027.77,510.368 2033.73,502.181 \n 2038.34,493.124 2041.49,483.426 2043.1,473.336 2043.12,463.111 2041.55,453.014 2038.43,443.306 2033.83,434.239 2027.89,426.047 2020.74,418.945 2012.57,413.119 \n 2003.6,408.724 1994.05,405.874 1984.19,404.648 1974.26,405.079 1964.53,407.159 1955.25,410.833 1946.68,416.007 1939.04,422.545 1932.54,430.275 1927.34,438.992 \n 1923.6,448.463 1921.41,458.436 1920.84,468.643 1921.89,478.81 1924.55,488.661 1928.75,497.93 1934.36,506.364 1941.25,513.733 1949.23,519.834 1958.07,524.499 \n 1967.54,527.598 1977.38,529.045 1987.31,528.798 1997.07,526.861 2006.38,523.286 2014.98,518.171 2022.64,511.655 2029.13,503.919 2034.29,495.177 2037.95,485.673 \n 2040.02,475.673 2040.44,465.458 2039.18,455.316 2036.3,445.533 2031.85,436.388 2025.98,428.14 2018.85,421.026 2010.65,415.249 2001.62,410.978 1992.02,408.336 \n 1982.13,407.401 1972.22,408.202 1962.59,410.717 1953.51,414.877 1945.25,420.561 1938.05,427.607 1932.12,435.811 1927.64,444.935 1924.73,454.711 1923.49,464.854 \n 1923.96,475.067 1926.11,485.047 1929.9,494.499 1935.2,503.144 1941.87,510.722 1949.71,517.008 1958.48,521.812 1967.92,524.989 1977.76,526.441 1987.69,526.124 \n 1997.42,524.045 2006.65,520.265 2015.1,514.895 2022.52,508.097 2028.68,500.077 2033.4,491.076 2036.51,481.369 2037.93,471.251 2037.62,461.034 2035.58,451.029 \n 2031.86,441.547 2026.59,432.881 2019.93,425.299 2012.08,419.038 2003.27,414.294 1993.8,411.216 1983.95,409.903 1974.02,410.398 1964.34,412.687 1955.2,416.7 \n 1946.9,422.312 1939.69,429.349 1933.81,437.589 1929.44,446.771 1926.73,456.606 1925.77,466.779 1926.57,476.968 1929.13,486.845 1933.37,496.093 1939.13,504.415 \n 1946.26,511.541 1954.51,517.24 1963.61,521.326 1973.28,523.664 1983.21,524.177 1993.06,522.845 2002.51,519.711 2011.27,514.875 2019.03,508.493 2025.54,500.775 \n 2030.59,491.972 2034.01,482.374 2035.68,472.297 2035.55,462.077 2033.61,452.051 2029.94,442.554 2024.64,433.903 2017.91,426.39 2009.95,420.266 2001.05,415.739 \n 1991.49,412.964 1981.59,412.035 1971.7,412.987 1962.15,415.789 1953.25,420.347 1945.33,426.508 1938.64,434.065 1933.42,442.76 1929.84,452.296 1928.04,462.348 \n 1928.07,472.569 1929.94,482.608 1933.58,492.117 1938.87,500.767 1945.63,508.256 1953.63,514.324 1962.58,518.756 1972.17,521.397 1982.08,522.153 1991.95,520.994 \n 2001.44,517.959 2010.21,513.154 2017.94,506.746 2024.38,498.96 2029.28,490.071 2032.48,480.393 2033.85,470.27 2033.34,460.062 2030.98,450.135 2026.84,440.844 \n 2021.07,432.524 2013.88,425.474 2005.52,419.951 1996.3,416.154 1986.54,414.225 1976.61,414.233 1966.86,416.182 1957.64,420.002 1949.3,425.555 1942.15,432.641 \n 1936.43,441.001 1932.37,450.329 1930.12,460.283 1929.76,470.496 1931.3,480.592 1934.7,490.196 1939.83,498.951 1946.49,506.53 1954.44,512.651 1963.39,517.083 \n 1973.01,519.659 1982.92,520.279 1992.77,518.919 2002.17,515.628 2010.78,510.529 2018.26,503.811 2024.34,495.728 2028.78,486.586 2031.41,476.731 2032.13,466.538 \n 2030.91,456.396 2027.79,446.694 2022.89,437.803 2016.4,430.067 2008.56,423.784 1999.69,419.198 1990.11,416.489 1980.2,415.762 1970.35,417.049 1960.93,420.301 \n 1952.32,425.394 1944.85,432.13 1938.82,440.247 1934.46,449.428 1931.94,459.315 1931.38,469.517 1932.78,479.633 1936.1,489.264 1941.21,498.026 1947.91,505.572 \n 1955.93,511.6 \n \"/>\n<circle clip-path=\"url(#clip9303)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1981.33\" cy=\"467.6\" r=\"18\"/>\n<circle clip-path=\"url(#clip9303)\" style=\"fill:#e26f46; stroke:none; fill-opacity:1\" cx=\"1981.33\" cy=\"467.6\" r=\"14\"/>\n<polygon clip-path=\"url(#clip9300)\" points=\"\n157.898,2309.04 1152.76,2309.04 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750.163,1867.77 \n 760.093,1868.37 770.04,1868.45 779.979,1868.02 789.884,1867.08 799.73,1865.62 809.491,1863.64 819.142,1861.16 828.657,1858.18 838.01,1854.69 847.176,1850.72 \n 856.129,1846.26 864.845,1841.32 873.3,1835.93 881.467,1830.09 889.326,1823.81 896.851,1817.12 904.021,1810.03 910.814,1802.55 917.21,1794.71 923.188,1786.53 \n 928.73,1778.03 933.819,1769.24 938.436,1760.17 942.568,1750.86 946.2,1741.33 949.319,1731.62 951.914,1721.74 953.976,1711.72 955.495,1701.61 956.466,1691.42 \n 956.883,1681.2 956.743,1670.96 956.045,1660.75 954.789,1650.6 952.976,1640.54 950.611,1630.6 947.699,1620.81 944.249,1611.21 940.269,1601.83 935.77,1592.7 \n 930.767,1583.86 925.274,1575.33 919.308,1567.14 912.887,1559.32 906.032,1551.9 898.766,1544.91 891.112,1538.38 883.095,1532.32 874.743,1526.76 866.084,1521.72 \n 857.147,1517.23 847.965,1513.29 838.57,1509.93 828.994,1507.16 819.273,1504.99 809.442,1503.43 799.537,1502.5 789.594,1502.18 779.652,1502.5 769.747,1503.45 \n 759.918,1505.02 750.203,1507.21 740.639,1510.03 731.265,1513.45 722.116,1517.47 713.231,1522.07 704.645,1527.23 696.393,1532.95 688.51,1539.19 681.029,1545.93 \n 673.98,1553.16 667.395,1560.83 661.301,1568.91 655.726,1577.39 650.695,1586.22 646.23,1595.36 642.351,1604.79 639.078,1614.45 636.426,1624.32 634.407,1634.34 \n 633.033,1644.47 632.313,1654.68 632.249,1664.91 632.846,1675.13 634.103,1685.28 636.015,1695.32 638.577,1705.21 641.778,1714.9 645.606,1724.35 650.046,1733.51 \n 655.078,1742.33 660.681,1750.79 666.832,1758.83 673.503,1766.42 680.663,1773.53 688.281,1780.11 696.322,1786.13 704.749,1791.57 713.522,1796.39 722.599,1800.57 \n 731.938,1804.1 741.494,1806.94 751.219,1809.08 761.068,1810.52 770.99,1811.24 780.936,1811.23 790.857,1810.5 800.702,1809.04 810.421,1806.86 819.964,1803.98 \n 829.281,1800.39 838.323,1796.13 847.043,1791.21 855.394,1785.65 863.331,1779.48 870.811,1772.74 877.793,1765.45 884.238,1757.65 890.11,1749.39 895.376,1740.71 \n 900.006,1731.65 903.972,1722.27 907.25,1712.61 909.822,1702.72 911.671,1692.67 912.783,1682.5 913.153,1672.27 912.774,1662.04 911.648,1651.88 909.779,1641.83 \n 907.175,1631.95 903.85,1622.3 899.821,1612.95 895.11,1603.94 889.742,1595.32 883.747,1587.15 877.16,1579.49 870.018,1572.37 862.362,1565.83 854.237,1559.93 \n 845.69,1554.7 836.773,1550.16 827.539,1546.36 818.045,1543.31 808.346,1541.04 798.505,1539.57 788.58,1538.89 778.635,1539.03 768.732,1539.97 758.933,1541.73 \n 749.302,1544.28 739.899,1547.62 730.786,1551.72 722.023,1556.56 713.667,1562.11 705.774,1568.33 698.397,1575.2 691.584,1582.65 685.384,1590.65 679.838,1599.15 \n 674.985,1608.08 670.859,1617.39 667.491,1627.02 664.904,1636.9 663.118,1646.97 662.148,1657.15 662.001,1667.38 662.682,1677.59 664.186,1687.71 666.507,1697.66 \n 669.628,1707.37 673.531,1716.79 678.19,1725.83 683.573,1734.43 689.643,1742.54 696.358,1750.08 703.672,1757.02 711.533,1763.29 719.885,1768.84 728.668,1773.64 \n 737.819,1777.65 747.271,1780.84 756.953,1783.17 766.796,1784.64 776.725,1785.23 786.666,1784.93 796.544,1783.74 806.284,1781.67 815.811,1778.73 825.052,1774.95 \n 833.935,1770.35 842.39,1764.96 850.351,1758.82 857.755,1751.99 864.542,1744.51 870.658,1736.44 876.053,1727.85 880.682,1718.79 884.506,1709.35 887.494,1699.59 \n 889.619,1689.59 890.862,1679.44 891.211,1669.21 890.661,1658.99 889.216,1648.87 886.884,1638.92 883.683,1629.24 879.638,1619.89 874.78,1610.96 869.15,1602.52 \n 862.791,1594.66 855.757,1587.42 848.105,1580.89 839.9,1575.11 831.211,1570.13 822.11,1566 812.677,1562.76 802.991,1560.44 793.137,1559.06 783.201,1558.63 \n 773.27,1559.16 763.43,1560.65 753.771,1563.08 744.377,1566.44 735.334,1570.7 726.723,1575.82 718.622,1581.75 711.107,1588.45 704.246,1595.86 698.104,1603.91 \n 692.739,1612.52 688.2,1621.63 684.532,1631.14 681.772,1640.97 679.945,1651.03 679.072,1661.22 679.162,1671.45 680.217,1681.62 682.229,1691.64 685.18,1701.41 \n 689.044,1710.84 693.787,1719.84 699.363,1728.31 705.721,1736.17 712.802,1743.36 720.536,1749.79 728.85,1755.4 737.664,1760.14 746.891,1763.96 756.442,1766.81 \n 766.221,1768.67 776.131,1769.51 786.074,1769.34 795.949,1768.13 805.657,1765.91 815.099,1762.7 824.178,1758.53 832.8,1753.43 840.877,1747.46 848.324,1740.68 \n 855.062,1733.16 861.021,1724.97 866.136,1716.19 870.353,1706.93 873.625,1697.26 875.916,1687.31 877.2,1677.16 877.462,1666.93 876.696,1656.73 874.909,1646.67 \n 872.118,1636.85 868.351,1627.38 863.648,1618.36 858.057,1609.9 851.638,1602.09 844.459,1595.01 836.599,1588.74 828.143,1583.36 819.184,1578.92 809.819,1575.48 \n 800.154,1573.07 790.296,1571.73 780.355,1571.48 770.444,1572.31 760.675,1574.22 751.16,1577.19 742.008,1581.19 733.324,1586.18 725.209,1592.09 717.759,1598.87 \n 711.061,1606.43 705.194,1614.69 700.228,1623.55 696.223,1632.92 693.229,1642.67 691.282,1652.7 690.407,1662.9 690.617,1673.12 691.911,1683.27 694.275,1693.21 \n 697.684,1702.82 702.098,1711.98 707.464,1720.6 713.721,1728.55 720.792,1735.74 728.592,1742.09 737.028,1747.5 745.996,1751.92 755.386,1755.29 765.081,1757.56 \n 774.963,1758.7 784.906,1758.7 794.787,1757.55 804.48,1755.27 813.863,1751.89 822.816,1747.44 831.224,1741.97 838.979,1735.57 845.979,1728.31 852.134,1720.27 \n 857.363,1711.57 861.595,1702.31 864.775,1692.62 866.858,1682.62 867.816,1672.43 867.634,1662.2 866.312,1652.06 863.867,1642.15 860.328,1632.59 855.74,1623.51 \n 850.165,1615.04 843.674,1607.29 836.353,1600.37 828.301,1594.37 819.625,1589.37 810.441,1585.45 800.876,1582.66 791.059,1581.03 781.125,1580.6 771.211,1581.38 \n 761.453,1583.35 751.989,1586.48 742.95,1590.74 734.464,1596.07 726.651,1602.4 719.621,1609.63 713.476,1617.68 708.303,1626.41 704.18,1635.72 701.165,1645.47 \n 699.304,1655.52 698.625,1665.73 699.141,1675.94 700.846,1686.02 703.717,1695.81 707.714,1705.18 712.781,1713.98 718.844,1722.09 725.818,1729.38 733.599,1735.75 \n 742.075,1741.1 751.12,1745.34 760.601,1748.42 770.377,1750.29 780.301,1750.91 790.224,1750.28 799.997,1748.4 809.471,1745.3 818.502,1741.02 826.951,1735.63 \n 834.688,1729.2 841.594,1721.84 847.561,1713.66 852.495,1704.78 856.318,1695.34 858.968,1685.48 860.404,1675.36 860.599,1665.13 859.55,1654.96 857.271,1645 \n 853.795,1635.42 849.177,1626.36 843.487,1617.97 836.814,1610.39 829.265,1603.73 820.958,1598.11 812.026,1593.62 802.613,1590.33 792.869,1588.29 782.953,1587.55 \n 773.027,1588.12 763.252,1589.98 753.788,1593.12 744.793,1597.48 736.415,1602.98 728.794,1609.55 722.058,1617.07 716.32,1625.43 711.678,1634.47 708.21,1644.06 \n 705.978,1654.03 705.02,1664.21 705.356,1674.43 706.98,1684.52 709.868,1694.31 713.972,1703.63 719.223,1712.31 725.534,1720.21 732.797,1727.2 740.888,1733.14 \n 749.667,1737.94 758.984,1741.51 768.677,1743.78 778.577,1744.72 788.51,1744.3 798.301,1742.53 807.778,1739.44 816.773,1735.09 825.125,1729.54 832.683,1722.89 \n 839.313,1715.27 844.894,1706.81 849.324,1697.65 852.522,1687.97 854.427,1677.93 855.004,1667.72 854.24,1657.52 852.147,1647.52 848.761,1637.9 844.143,1628.85 \n 838.375,1620.52 831.563,1613.07 823.831,1606.64 815.322,1601.35 806.191,1597.31 796.61,1594.58 786.757,1593.22 776.817,1593.27 766.976,1594.71 757.42,1597.53 \n 748.332,1601.68 739.884,1607.07 732.239,1613.6 725.543,1621.16 719.926,1629.6 715.498,1638.76 712.346,1648.46 710.533,1658.52 710.095,1668.73 711.043,1678.91 \n 713.361,1688.86 717.005,1698.38 721.906,1707.27 727.969,1715.38 735.076,1722.53 743.088,1728.58 751.847,1733.42 761.18,1736.94 770.902,1739.07 780.82,1739.77 \n 790.733,1739.01 800.443,1736.83 809.754,1733.25 818.476,1728.34 826.433,1722.21 833.461,1714.98 839.415,1706.79 844.172,1697.81 847.633,1688.22 849.725,1678.22 \n 850.403,1668.02 849.651,1657.82 847.482,1647.84 843.94,1638.29 839.096,1629.36 833.051,1621.24 825.929,1614.1 817.879,1608.1 809.069,1603.37 799.685,1600 \n 789.924,1598.07 779.994,1597.62 770.106,1598.66 760.472,1601.17 751.298,1605.11 742.783,1610.39 735.111,1616.89 728.448,1624.48 722.941,1632.99 718.71,1642.24 \n 715.851,1652.04 714.426,1662.16 714.469,1672.39 715.982,1682.49 718.932,1692.26 723.257,1701.47 728.862,1709.91 735.625,1717.41 743.397,1723.78 752.005,1728.89 \n 761.259,1732.62 770.951,1734.89 780.863,1735.64 790.773,1734.86 800.456,1732.55 809.692,1728.77 818.271,1723.61 825.996,1717.17 832.689,1709.61 838.196,1701.1 \n 842.388,1691.83 845.167,1682.01 846.468,1671.87 846.257,1661.65 844.539,1651.58 841.351,1641.89 836.765,1632.82 830.887,1624.57 823.854,1617.35 815.829,1611.31 \n 807.001,1606.62 797.577,1603.37 787.78,1601.65 777.842,1601.5 768.001,1602.93 758.491,1605.9 749.541,1610.34 741.366,1616.16 734.165,1623.21 728.112,1631.32 \n 723.356,1640.3 720.013,1649.92 718.167,1659.97 717.865,1670.19 719.115,1680.34 721.89,1690.16 726.123,1699.41 731.712,1707.86 738.519,1715.31 746.379,1721.57 \n 755.096,1726.48 764.456,1729.92 774.226,1731.8 784.16,1732.06 794.012,1730.71 803.531,1727.78 812.479,1723.33 820.629,1717.48 827.772,1710.37 833.726,1702.18 \n 838.338,1693.12 841.489,1683.43 843.096,1673.34 843.116,1663.11 841.547,1653.01 838.426,1643.31 833.834,1634.24 827.886,1626.05 820.737,1618.95 812.57,1613.12 \n 803.597,1608.72 794.053,1605.87 784.187,1604.65 774.259,1605.08 764.529,1607.16 755.255,1610.83 746.684,1616.01 739.043,1622.55 732.539,1630.27 727.344,1638.99 \n 723.601,1648.46 721.411,1658.44 720.835,1668.64 721.89,1678.81 724.55,1688.66 728.745,1697.93 734.363,1706.36 741.252,1713.73 749.226,1719.83 758.068,1724.5 \n 767.538,1727.6 777.375,1729.05 787.309,1728.8 797.066,1726.86 806.376,1723.29 814.98,1718.17 822.638,1711.65 829.134,1703.92 834.288,1695.18 837.951,1685.67 \n 840.021,1675.67 840.436,1665.46 839.183,1655.32 836.296,1645.53 831.855,1636.39 825.983,1628.14 818.846,1621.03 810.647,1615.25 801.619,1610.98 792.02,1608.34 \n 782.125,1607.4 772.219,1608.2 762.588,1610.72 753.511,1614.88 745.252,1620.56 738.053,1627.61 732.124,1635.81 727.64,1644.93 724.734,1654.71 723.493,1664.85 \n 723.956,1675.07 726.111,1685.05 729.896,1694.5 735.201,1703.14 741.87,1710.72 749.706,1717.01 758.477,1721.81 767.921,1724.99 777.756,1726.44 787.688,1726.12 \n 797.416,1724.05 806.648,1720.26 815.103,1714.89 822.524,1708.1 828.684,1700.08 833.395,1691.08 836.511,1681.37 837.935,1671.25 837.621,1661.03 835.577,1651.03 \n 831.864,1641.55 826.594,1632.88 819.93,1625.3 812.076,1619.04 803.275,1614.29 793.8,1611.22 783.947,1609.9 774.023,1610.4 764.34,1612.69 755.202,1616.7 \n 746.898,1622.31 739.691,1629.35 733.81,1637.59 729.444,1646.77 726.733,1656.61 725.765,1666.78 726.573,1676.97 729.133,1686.84 733.365,1696.09 739.135,1704.42 \n 746.258,1711.54 754.506,1717.24 763.613,1721.33 773.285,1723.66 783.207,1724.18 793.058,1722.85 802.514,1719.71 811.267,1714.87 819.028,1708.49 825.541,1700.77 \n 830.591,1691.97 834.008,1682.37 835.679,1672.3 835.546,1662.08 833.611,1652.05 829.936,1642.55 824.644,1633.9 817.908,1626.39 809.954,1620.27 801.046,1615.74 \n 791.485,1612.96 781.592,1612.04 771.7,1612.99 762.146,1615.79 753.254,1620.35 745.328,1626.51 738.638,1634.07 733.416,1642.76 729.84,1652.3 728.037,1662.35 \n 728.069,1672.57 729.937,1682.61 733.58,1692.12 738.872,1700.77 745.633,1708.26 753.627,1714.32 762.578,1718.76 772.175,1721.4 782.082,1722.15 791.952,1720.99 \n 801.438,1717.96 810.205,1713.15 817.944,1706.75 824.38,1698.96 829.283,1690.07 832.477,1680.39 833.847,1670.27 833.342,1660.06 830.978,1650.14 826.839,1640.84 \n 821.07,1632.52 813.878,1625.47 805.52,1619.95 796.297,1616.15 786.542,1614.22 776.608,1614.23 766.857,1616.18 757.644,1620 749.305,1625.56 742.146,1632.64 \n 736.432,1641 732.373,1650.33 730.12,1660.28 729.759,1670.5 731.305,1680.59 734.702,1690.2 739.827,1698.95 746.489,1706.53 754.444,1712.65 763.394,1717.08 \n 773.007,1719.66 782.921,1720.28 792.766,1718.92 802.17,1715.63 810.778,1710.53 818.263,1703.81 824.342,1695.73 828.781,1686.59 831.411,1676.73 832.129,1666.54 \n 830.906,1656.4 827.786,1646.69 822.888,1637.8 816.398,1630.07 808.565,1623.78 799.688,1619.2 790.111,1616.49 780.204,1615.76 770.351,1617.05 760.935,1620.3 \n 752.323,1625.39 744.854,1632.13 738.82,1640.25 734.459,1649.43 731.945,1659.31 731.378,1669.52 732.782,1679.63 736.104,1689.26 741.215,1698.03 747.912,1705.57 \n 755.932,1711.6 764.956,1715.87 774.625,1718.21 784.552,1718.52 794.34,1716.78 803.596,1713.08 811.947,1707.55 819.054,1700.41 824.631,1691.95 828.448,1682.52 \n 830.35,1672.49 830.258,1662.27 828.174,1652.28 824.179,1642.93 818.437,1634.59 811.181,1627.61 802.707,1622.28 793.361,1618.82 783.53,1617.38 773.617,1618.01 \n 764.034,1620.69 755.177,1625.31 747.415,1631.69 741.073,1639.55 736.416,1648.57 733.639,1658.39 732.862,1668.57 734.119,1678.71 737.359,1688.37 742.446,1697.14 \n 749.17,1704.66 757.246,1710.61 766.334,1714.73 776.05,1716.84 785.981,1716.86 795.704,1714.78 804.804,1710.69 812.892,1704.76 819.62,1697.25 824.697,1688.47 \n 827.905,1678.8 829.103,1668.65 828.237,1658.48 825.344,1648.7 820.545,1639.76 814.049,1632.03 806.136,1625.86 797.149,1621.51 787.48,1619.18 777.552,1618.97 \n 767.799,1620.89 758.65,1624.86 750.508,1630.71 743.732,1638.18 738.624,1646.94 735.411,1656.61 734.237,1666.76 735.156,1676.93 738.13,1686.68 743.027,1695.56 \n 749.63,1703.2 757.646,1709.23 766.717,1713.38 776.435,1715.48 786.365,1715.42 796.058,1713.2 805.077,1708.93 813.012,1702.79 819.504,1695.06 824.254,1686.08 \n 827.047,1676.28 827.753,1666.09 826.337,1655.98 822.862,1646.41 817.486,1637.82 810.454,1630.6 802.089,1625.1 792.774,1621.56 782.94,1620.15 773.04,1620.94 \n 763.534,1623.89 754.864,1628.87 747.434,1635.65 741.592,1643.91 737.611,1653.27 735.68,1663.29 735.892,1673.5 738.238,1683.43 742.609,1692.6 748.802,1700.59 \n 756.526,1707.01 765.416,1711.56 775.053,1714.02 784.978,1714.27 794.721,1712.31 803.817,1708.21 811.831,1702.18 818.379,1694.5 823.145,1685.54 825.899,1675.72 \n 826.507,1665.53 824.938,1655.44 821.265,1645.95 815.664,1637.52 808.406,1630.55 799.84,1625.38 790.382,1622.27 780.492,1621.38 770.652,1622.74 761.344,1626.29 \n 753.022,1631.86 746.098,1639.18 740.911,1647.89 737.72,1657.56 736.682,1667.72 737.852,1677.87 741.174,1687.49 746.483,1696.12 753.52,1703.33 761.934,1708.75 \n 771.308,1712.11 781.174,1713.25 791.039,1712.11 800.408,1708.73 808.811,1703.29 815.825,1696.06 821.094,1687.4 824.351,1677.76 825.429,1667.6 824.273,1657.46 \n 820.939,1647.84 815.594,1639.23 808.509,1632.07 800.044,1626.74 790.63,1623.49 780.749,1622.51 770.906,1623.84 761.608,1627.42 753.334,1633.07 746.51,1640.48 \n 741.491,1649.3 738.538,1659.05 737.806,1669.23 739.334,1679.33 743.045,1688.8 748.747,1697.16 756.145,1703.97 764.854,1708.87 774.418,1711.61 784.336,1712.03 \n 794.087,1710.12 803.159,1705.98 811.071,1699.81 817.405,1691.95 821.825,1682.8 824.094,1672.86 824.091,1662.64 821.813,1652.7 817.38,1643.57 811.027,1635.72 \n 803.091,1629.58 793.998,1625.49 784.232,1623.66 774.32,1624.19 764.793,1627.06 756.167,1632.11 748.908,1639.08 743.41,1647.58 739.972,1657.16 738.783,1667.3 \n 739.91,1677.45 743.293,1687.05 748.749,1695.58 755.981,1702.57 764.596,1707.65 774.121,1710.52 784.034,1711.03 793.79,1709.15 802.851,1704.98 810.715,1698.75 \n 816.947,1690.8 821.2,1681.57 823.237,1671.58 822.942,1661.37 820.329,1651.52 815.543,1642.57 808.849,1635.03 800.621,1629.32 791.318,1625.76 781.463,1624.56 \n 771.608,1625.77 762.31,1629.35 754.094,1635.07 747.425,1642.64 742.682,1651.61 740.136,1661.48 739.933,1671.69 742.087,1681.65 746.477,1690.81 752.854,1698.64 \n 760.854,1704.68 770.021,1708.59 779.83,1710.15 789.717,1709.26 799.112,1705.97 807.474,1700.47 814.319,1693.08 819.249,1684.21 821.976,1674.39 822.342,1664.19 \n 820.322,1654.19 816.032,1644.99 809.721,1637.11 801.756,1631.01 792.603,1627.07 782.796,1625.5 772.911,1626.41 763.53,1629.75 755.206,1635.31 748.431,1642.77 \n 743.607,1651.69 741.021,1661.55 740.829,1671.76 743.043,1681.71 747.533,1690.81 754.035,1698.53 762.161,1704.39 771.426,1708.05 781.278,1709.28 791.124,1708.01 \n 800.374,1704.31 808.47,1698.4 814.923,1690.65 819.343,1681.51 821.46,1671.53 821.145,1661.33 818.414,1651.51 813.432,1642.68 806.5,1635.37 798.039,1630.04 \n 788.565,1627 778.657,1626.45 768.92,1628.42 759.953,1632.79 752.308,1639.3 746.455,1647.55 742.758,1657.02 741.446,1667.14 742.603,1677.28 746.157,1686.81 \n 751.892,1695.15 759.452,1701.76 768.368,1706.24 778.084,1708.31 787.996,1707.83 797.483,1704.84 805.951,1699.52 812.866,1692.2 817.795,1683.34 820.424,1673.49 \n 820.587,1663.29 818.271,1653.36 813.62,1644.34 806.928,1636.8 798.618,1631.23 789.216,1627.96 779.32,1627.23 769.56,1629.07 760.559,1633.36 752.895,1639.84 \n 747.057,1648.1 743.423,1657.6 742.228,1667.73 743.55,1677.85 747.305,1687.3 753.254,1695.47 761.014,1701.83 770.083,1705.97 779.874,1707.62 789.753,1706.67 \n 799.075,1703.17 807.234,1697.36 813.697,1689.62 818.039,1680.44 819.975,1670.43 819.376,1660.24 816.279,1650.54 810.887,1641.97 803.552,1635.1 794.758,1630.38 \n 785.083,1628.11 775.168,1628.47 765.669,1631.41 757.218,1636.76 750.376,1644.15 745.601,1653.1 743.212,1663.01 743.371,1673.21 746.069,1683.03 751.127,1691.82 \n 758.208,1698.96 766.837,1704 776.435,1706.58 786.355,1706.54 795.929,1703.86 804.508,1698.74 811.511,1691.51 816.463,1682.66 819.024,1672.8 819.02,1662.59 \n 816.448,1652.74 811.483,1643.9 804.461,1636.69 795.862,1631.6 786.274,1628.98 776.354,1629.02 766.784,1631.71 758.223,1636.86 751.262,1644.13 746.384,1653.02 \n 743.926,1662.91 744.062,1673.12 746.782,1682.93 751.901,1691.67 759.064,1698.73 767.774,1703.62 777.422,1705.99 787.337,1705.67 796.824,1702.69 805.217,1697.25 \n 811.927,1689.73 816.481,1680.67 818.556,1670.69 818.004,1660.5 814.863,1650.82 809.352,1642.33 801.861,1635.64 792.919,1631.22 783.16,1629.4 773.279,1630.29 \n 763.98,1633.84 755.926,1639.79 749.694,1647.73 745.732,1657.09 744.324,1667.19 745.575,1677.31 749.396,1686.73 755.514,1694.76 763.489,1700.83 772.746,1704.5 \n 782.618,1705.49 792.389,1703.73 801.35,1699.36 808.851,1692.68 814.344,1684.19 817.427,1674.49 817.875,1664.29 815.652,1654.35 810.919,1645.38 804.021,1638.05 \n 795.463,1632.89 785.874,1630.29 775.957,1630.43 766.444,1633.32 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fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1333.9, 2121.82)\" x=\"1333.9\" y=\"2121.82\">0.2</text>\n</g>\n<g clip-path=\"url(#clip9300)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1333.9, 1917.11)\" x=\"1333.9\" y=\"1917.11\">0.4</text>\n</g>\n<g clip-path=\"url(#clip9300)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1333.9, 1712.39)\" x=\"1333.9\" y=\"1712.39\">0.6</text>\n</g>\n<g clip-path=\"url(#clip9300)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1333.9, 1507.67)\" x=\"1333.9\" y=\"1507.67\">0.8</text>\n</g>\n<g clip-path=\"url(#clip9300)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1333.9, 1302.95)\" x=\"1333.9\" y=\"1302.95\">1.0</text>\n</g>\n<g clip-path=\"url(#clip9300)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 1855.33, 1251.6)\" x=\"1855.33\" y=\"1251.6\">Euler spiral (t = 0 to 40)</text>\n</g>\n<polyline clip-path=\"url(#clip9305)\" style=\"stroke:#009af9; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n 1357.9,2309.04 1367.85,2309.04 1377.8,2309.04 1387.74,2309.03 1397.69,2309.02 1407.64,2309 1417.59,2308.97 1427.54,2308.93 1437.49,2308.87 1447.43,2308.79 \n 1457.38,2308.7 1467.33,2308.59 1477.28,2308.45 1487.23,2308.29 1497.17,2308.11 1507.12,2307.89 1517.06,2307.65 1527.01,2307.37 1536.95,2307.05 1546.9,2306.7 \n 1556.84,2306.31 1566.78,2305.88 1576.72,2305.41 1586.65,2304.89 1596.58,2304.33 1606.52,2303.71 1616.44,2303.05 1626.37,2302.33 1636.29,2301.56 1646.2,2300.73 \n 1656.11,2299.84 1666.02,2298.89 1675.92,2297.87 1685.81,2296.79 1695.7,2295.65 1705.58,2294.43 1715.45,2293.14 1725.31,2291.78 1735.16,2290.35 1745,2288.84 \n 1754.82,2287.25 1764.64,2285.58 1774.44,2283.82 1784.23,2281.98 1794,2280.06 1803.75,2278.04 1813.49,2275.94 1823.2,2273.74 1832.9,2271.45 1842.58,2269.07 \n 1852.23,2266.58 1861.85,2264 1871.45,2261.32 1881.03,2258.53 1890.57,2255.64 1900.08,2252.65 1909.56,2249.54 1919.01,2246.33 1928.42,2243.01 1937.79,2239.57 \n 1947.12,2236.02 1956.41,2232.36 1965.66,2228.58 1974.86,2224.68 1984.01,2220.67 1993.11,2216.53 2002.15,2212.27 2011.15,2207.89 2020.08,2203.39 2028.95,2198.76 \n 2037.76,2194 2046.51,2189.12 2055.18,2184.12 2063.79,2178.98 2072.32,2173.72 2080.78,2168.32 2089.15,2162.8 2097.44,2157.14 2105.65,2151.36 2113.77,2145.44 \n 2121.8,2139.39 2129.73,2133.22 2137.56,2126.91 2145.29,2120.47 2152.92,2113.89 2160.44,2107.19 2167.85,2100.36 2175.14,2093.39 2182.31,2086.3 2189.36,2079.08 \n 2196.28,2071.73 2203.08,2064.25 2209.74,2056.65 2216.26,2048.92 2222.64,2041.07 2228.88,2033.09 2234.97,2025 2240.91,2016.79 2246.69,2008.46 2252.31,2000.01 \n 2257.77,1991.45 2263.06,1982.79 2268.18,1974.01 2273.13,1965.13 2277.89,1956.14 2282.47,1947.06 2286.87,1937.88 2291.08,1928.6 2295.09,1919.23 2298.91,1909.78 \n 2302.52,1900.24 2305.93,1890.63 2309.13,1880.94 2312.12,1871.17 2314.89,1861.34 2317.45,1851.45 2319.78,1841.5 2321.89,1831.5 2323.77,1821.45 2325.42,1811.35 \n 2326.83,1801.22 2328.01,1791.06 2328.95,1780.87 2329.65,1770.66 2330.1,1760.43 2330.31,1750.2 2330.27,1739.96 2329.97,1729.73 2329.43,1719.51 2328.63,1709.31 \n 2327.58,1699.13 2326.26,1688.98 2324.69,1678.88 2322.87,1668.82 2320.78,1658.81 2318.43,1648.86 2315.82,1638.98 2312.95,1629.18 2309.83,1619.47 2306.44,1609.84 \n 2302.79,1600.32 2298.89,1590.9 2294.73,1581.61 2290.31,1572.43 2285.65,1563.4 2280.73,1554.5 2275.56,1545.75 2270.15,1537.17 2264.49,1528.75 2258.59,1520.5 \n 2252.46,1512.44 2246.09,1504.58 2239.5,1496.91 2232.69,1489.46 2225.65,1482.22 2218.4,1475.21 2210.95,1468.43 2203.29,1461.9 2195.44,1455.61 2187.39,1449.59 \n 2179.17,1443.83 2170.77,1438.34 2162.2,1433.14 2153.48,1428.22 2144.6,1423.6 2135.58,1419.29 2126.43,1415.28 2117.15,1411.58 2107.76,1408.21 2098.26,1405.16 \n 2088.67,1402.44 2079,1400.06 2069.25,1398.03 2059.44,1396.34 2049.57,1394.99 2039.67,1394.01 2029.74,1393.38 2019.8,1393.11 2009.85,1393.2 1999.91,1393.66 \n 1990,1394.48 1980.12,1395.67 1970.28,1397.23 1960.51,1399.15 1950.82,1401.44 1941.21,1404.09 1931.7,1407.11 1922.31,1410.49 1913.05,1414.23 1903.94,1418.33 \n 1894.98,1422.78 1886.19,1427.57 1877.58,1432.71 1869.18,1438.18 1860.98,1443.98 1853.01,1450.1 1845.28,1456.54 1837.8,1463.29 1830.58,1470.33 1823.64,1477.67 \n 1816.99,1485.28 1810.64,1493.16 1804.6,1501.29 1798.89,1509.67 1793.51,1518.28 1788.48,1527.11 1783.8,1536.14 1779.49,1545.37 1775.55,1554.77 1772,1564.33 \n 1768.84,1574.03 1766.08,1583.87 1763.72,1593.81 1761.78,1603.85 1760.25,1613.96 1759.14,1624.13 1758.46,1634.34 1758.21,1644.58 1758.4,1654.81 1759.01,1665.03 \n 1760.06,1675.2 1761.54,1685.32 1763.46,1695.37 1765.81,1705.31 1768.58,1715.14 1771.78,1724.83 1775.41,1734.37 1779.44,1743.72 1783.89,1752.88 1788.73,1761.81 \n 1793.97,1770.52 1799.6,1778.96 1805.59,1787.13 1811.95,1795 1818.65,1802.56 1825.7,1809.79 1833.06,1816.67 1840.73,1823.18 1848.7,1829.31 1856.94,1835.05 \n 1865.43,1840.38 1874.17,1845.27 1883.12,1849.74 1892.27,1853.75 1901.6,1857.29 1911.09,1860.37 1920.71,1862.96 1930.45,1865.06 1940.27,1866.67 1950.16,1867.77 \n 1960.09,1868.37 1970.04,1868.45 1979.98,1868.02 1989.88,1867.08 1999.73,1865.62 2009.49,1863.64 2019.14,1861.16 2028.66,1858.18 2038.01,1854.69 2047.18,1850.72 \n 2056.13,1846.26 2064.85,1841.32 2073.3,1835.93 2081.47,1830.09 2089.33,1823.81 2096.85,1817.12 2104.02,1810.03 2110.81,1802.55 2117.21,1794.71 2123.19,1786.53 \n 2128.73,1778.03 2133.82,1769.24 2138.44,1760.17 2142.57,1750.86 2146.2,1741.33 2149.32,1731.62 2151.91,1721.74 2153.98,1711.72 2155.5,1701.61 2156.47,1691.42 \n 2156.88,1681.2 2156.74,1670.96 2156.05,1660.75 2154.79,1650.6 2152.98,1640.54 2150.61,1630.6 2147.7,1620.81 2144.25,1611.21 2140.27,1601.83 2135.77,1592.7 \n 2130.77,1583.86 2125.27,1575.33 2119.31,1567.14 2112.89,1559.32 2106.03,1551.9 2098.77,1544.91 2091.11,1538.38 2083.09,1532.32 2074.74,1526.76 2066.08,1521.72 \n 2057.15,1517.23 2047.97,1513.29 2038.57,1509.93 2028.99,1507.16 2019.27,1504.99 2009.44,1503.43 1999.54,1502.5 1989.59,1502.18 1979.65,1502.5 1969.75,1503.45 \n 1959.92,1505.02 1950.2,1507.21 1940.64,1510.03 1931.26,1513.45 1922.12,1517.47 1913.23,1522.07 1904.65,1527.23 1896.39,1532.95 1888.51,1539.19 1881.03,1545.93 \n 1873.98,1553.16 1867.39,1560.83 1861.3,1568.91 1855.73,1577.39 1850.7,1586.22 1846.23,1595.36 1842.35,1604.79 1839.08,1614.45 1836.43,1624.32 1834.41,1634.34 \n 1833.03,1644.47 1832.31,1654.68 1832.25,1664.91 1832.85,1675.13 1834.1,1685.28 1836.02,1695.32 1838.58,1705.21 1841.78,1714.9 1845.61,1724.35 1850.05,1733.51 \n 1855.08,1742.33 1860.68,1750.79 1866.83,1758.83 1873.5,1766.42 1880.66,1773.53 1888.28,1780.11 1896.32,1786.13 1904.75,1791.57 1913.52,1796.39 1922.6,1800.57 \n 1931.94,1804.1 1941.49,1806.94 1951.22,1809.08 1961.07,1810.52 1970.99,1811.24 1980.94,1811.23 1990.86,1810.5 2000.7,1809.04 2010.42,1806.86 2019.96,1803.98 \n 2029.28,1800.39 2038.32,1796.13 2047.04,1791.21 2055.39,1785.65 2063.33,1779.48 2070.81,1772.74 2077.79,1765.45 2084.24,1757.65 2090.11,1749.39 2095.38,1740.71 \n 2100.01,1731.65 2103.97,1722.27 2107.25,1712.61 2109.82,1702.72 2111.67,1692.67 2112.78,1682.5 2113.15,1672.27 2112.77,1662.04 2111.65,1651.88 2109.78,1641.83 \n 2107.18,1631.95 2103.85,1622.3 2099.82,1612.95 2095.11,1603.94 2089.74,1595.32 2083.75,1587.15 2077.16,1579.49 2070.02,1572.37 2062.36,1565.83 2054.24,1559.93 \n 2045.69,1554.7 2036.77,1550.16 2027.54,1546.36 2018.04,1543.31 2008.35,1541.04 1998.5,1539.57 1988.58,1538.89 1978.64,1539.03 1968.73,1539.97 1958.93,1541.73 \n 1949.3,1544.28 1939.9,1547.62 1930.79,1551.72 1922.02,1556.56 1913.67,1562.11 1905.77,1568.33 1898.4,1575.2 1891.58,1582.65 1885.38,1590.65 1879.84,1599.15 \n 1874.98,1608.08 1870.86,1617.39 1867.49,1627.02 1864.9,1636.9 1863.12,1646.97 1862.15,1657.15 1862,1667.38 1862.68,1677.59 1864.19,1687.71 1866.51,1697.66 \n 1869.63,1707.37 1873.53,1716.79 1878.19,1725.83 1883.57,1734.43 1889.64,1742.54 1896.36,1750.08 1903.67,1757.02 1911.53,1763.29 1919.89,1768.84 1928.67,1773.64 \n 1937.82,1777.65 1947.27,1780.84 1956.95,1783.17 1966.8,1784.64 1976.73,1785.23 1986.67,1784.93 1996.54,1783.74 2006.28,1781.67 2015.81,1778.73 2025.05,1774.95 \n 2033.93,1770.35 2042.39,1764.96 2050.35,1758.82 2057.76,1751.99 2064.54,1744.51 2070.66,1736.44 2076.05,1727.85 2080.68,1718.79 2084.51,1709.35 2087.49,1699.59 \n 2089.62,1689.59 2090.86,1679.44 2091.21,1669.21 2090.66,1658.99 2089.22,1648.87 2086.88,1638.92 2083.68,1629.24 2079.64,1619.89 2074.78,1610.96 2069.15,1602.52 \n 2062.79,1594.66 2055.76,1587.42 2048.11,1580.89 2039.9,1575.11 2031.21,1570.13 2022.11,1566 2012.68,1562.76 2002.99,1560.44 1993.14,1559.06 1983.2,1558.63 \n 1973.27,1559.16 1963.43,1560.65 1953.77,1563.08 1944.38,1566.44 1935.33,1570.7 1926.72,1575.82 1918.62,1581.75 1911.11,1588.45 1904.25,1595.86 1898.1,1603.91 \n 1892.74,1612.52 1888.2,1621.63 1884.53,1631.14 1881.77,1640.97 1879.95,1651.03 1879.07,1661.22 1879.16,1671.45 1880.22,1681.62 1882.23,1691.64 1885.18,1701.41 \n 1889.04,1710.84 1893.79,1719.84 1899.36,1728.31 1905.72,1736.17 1912.8,1743.36 1920.54,1749.79 1928.85,1755.4 1937.66,1760.14 1946.89,1763.96 1956.44,1766.81 \n 1966.22,1768.67 1976.13,1769.51 1986.07,1769.34 1995.95,1768.13 2005.66,1765.91 2015.1,1762.7 2024.18,1758.53 2032.8,1753.43 2040.88,1747.46 2048.32,1740.68 \n 2055.06,1733.16 2061.02,1724.97 2066.14,1716.19 2070.35,1706.93 2073.62,1697.26 2075.92,1687.31 2077.2,1677.16 2077.46,1666.93 2076.7,1656.73 2074.91,1646.67 \n 2072.12,1636.85 2068.35,1627.38 2063.65,1618.36 2058.06,1609.9 2051.64,1602.09 2044.46,1595.01 2036.6,1588.74 2028.14,1583.36 2019.18,1578.92 2009.82,1575.48 \n 2000.15,1573.07 1990.3,1571.73 1980.35,1571.48 1970.44,1572.31 1960.68,1574.22 1951.16,1577.19 1942.01,1581.19 1933.32,1586.18 1925.21,1592.09 1917.76,1598.87 \n 1911.06,1606.43 1905.19,1614.69 1900.23,1623.55 1896.22,1632.92 1893.23,1642.67 1891.28,1652.7 1890.41,1662.9 1890.62,1673.12 1891.91,1683.27 1894.28,1693.21 \n 1897.68,1702.82 1902.1,1711.98 1907.46,1720.6 1913.72,1728.55 1920.79,1735.74 1928.59,1742.09 1937.03,1747.5 1946,1751.92 1955.39,1755.29 1965.08,1757.56 \n 1974.96,1758.7 1984.91,1758.7 1994.79,1757.55 2004.48,1755.27 2013.86,1751.89 2022.82,1747.44 2031.22,1741.97 2038.98,1735.57 2045.98,1728.31 2052.13,1720.27 \n 2057.36,1711.57 2061.6,1702.31 2064.77,1692.62 2066.86,1682.62 2067.82,1672.43 2067.63,1662.2 2066.31,1652.06 2063.87,1642.15 2060.33,1632.59 2055.74,1623.51 \n 2050.16,1615.04 2043.67,1607.29 2036.35,1600.37 2028.3,1594.37 2019.62,1589.37 2010.44,1585.45 2000.88,1582.66 1991.06,1581.03 1981.13,1580.6 1971.21,1581.38 \n 1961.45,1583.35 1951.99,1586.48 1942.95,1590.74 1934.46,1596.07 1926.65,1602.4 1919.62,1609.63 1913.48,1617.68 1908.3,1626.41 1904.18,1635.72 1901.16,1645.47 \n 1899.3,1655.52 1898.63,1665.73 1899.14,1675.94 1900.85,1686.02 1903.72,1695.81 1907.71,1705.18 1912.78,1713.98 1918.84,1722.09 1925.82,1729.38 1933.6,1735.75 \n 1942.08,1741.1 1951.12,1745.34 1960.6,1748.42 1970.38,1750.29 1980.3,1750.91 1990.22,1750.28 2000,1748.4 2009.47,1745.3 2018.5,1741.02 2026.95,1735.63 \n 2034.69,1729.2 2041.59,1721.84 2047.56,1713.66 2052.49,1704.78 2056.32,1695.34 2058.97,1685.48 2060.4,1675.36 2060.6,1665.13 2059.55,1654.96 2057.27,1645 \n 2053.8,1635.42 2049.18,1626.36 2043.49,1617.97 2036.81,1610.39 2029.26,1603.73 2020.96,1598.11 2012.03,1593.62 2002.61,1590.33 1992.87,1588.29 1982.95,1587.55 \n 1973.03,1588.12 1963.25,1589.98 1953.79,1593.12 1944.79,1597.48 1936.42,1602.98 1928.79,1609.55 1922.06,1617.07 1916.32,1625.43 1911.68,1634.47 1908.21,1644.06 \n 1905.98,1654.03 1905.02,1664.21 1905.36,1674.43 1906.98,1684.52 1909.87,1694.31 1913.97,1703.63 1919.22,1712.31 1925.53,1720.21 1932.8,1727.2 1940.89,1733.14 \n 1949.67,1737.94 1958.98,1741.51 1968.68,1743.78 1978.58,1744.72 1988.51,1744.3 1998.3,1742.53 2007.78,1739.44 2016.77,1735.09 2025.12,1729.54 2032.68,1722.89 \n 2039.31,1715.27 2044.89,1706.81 2049.32,1697.65 2052.52,1687.97 2054.43,1677.93 2055,1667.72 2054.24,1657.52 2052.15,1647.52 2048.76,1637.9 2044.14,1628.85 \n 2038.38,1620.52 2031.56,1613.07 2023.83,1606.64 2015.32,1601.35 2006.19,1597.31 1996.61,1594.58 1986.76,1593.22 1976.82,1593.27 1966.98,1594.71 1957.42,1597.53 \n 1948.33,1601.68 1939.88,1607.07 1932.24,1613.6 1925.54,1621.16 1919.93,1629.6 1915.5,1638.76 1912.35,1648.46 1910.53,1658.52 1910.1,1668.73 1911.04,1678.91 \n 1913.36,1688.86 1917.01,1698.38 1921.91,1707.27 1927.97,1715.38 1935.08,1722.53 1943.09,1728.58 1951.85,1733.42 1961.18,1736.94 1970.9,1739.07 1980.82,1739.77 \n 1990.73,1739.01 2000.44,1736.83 2009.75,1733.25 2018.48,1728.34 2026.43,1722.21 2033.46,1714.98 2039.41,1706.79 2044.17,1697.81 2047.63,1688.22 2049.73,1678.22 \n 2050.4,1668.02 2049.65,1657.82 2047.48,1647.84 2043.94,1638.29 2039.1,1629.36 2033.05,1621.24 2025.93,1614.1 2017.88,1608.1 2009.07,1603.37 1999.68,1600 \n 1989.92,1598.07 1979.99,1597.62 1970.11,1598.66 1960.47,1601.17 1951.3,1605.11 1942.78,1610.39 1935.11,1616.89 1928.45,1624.48 1922.94,1632.99 1918.71,1642.24 \n 1915.85,1652.04 1914.43,1662.16 1914.47,1672.39 1915.98,1682.49 1918.93,1692.26 1923.26,1701.47 1928.86,1709.91 1935.63,1717.41 1943.4,1723.78 1952.01,1728.89 \n 1961.26,1732.62 1970.95,1734.89 1980.86,1735.64 1990.77,1734.86 2000.46,1732.55 2009.69,1728.77 2018.27,1723.61 2026,1717.17 2032.69,1709.61 2038.2,1701.1 \n 2042.39,1691.83 2045.17,1682.01 2046.47,1671.87 2046.26,1661.65 2044.54,1651.58 2041.35,1641.89 2036.76,1632.82 2030.89,1624.57 2023.85,1617.35 2015.83,1611.31 \n 2007,1606.62 1997.58,1603.37 1987.78,1601.65 1977.84,1601.5 1968,1602.93 1958.49,1605.9 1949.54,1610.34 1941.37,1616.16 1934.16,1623.21 1928.11,1631.32 \n 1923.36,1640.3 1920.01,1649.92 1918.17,1659.97 1917.86,1670.19 1919.12,1680.34 1921.89,1690.16 1926.12,1699.41 1931.71,1707.86 1938.52,1715.31 1946.38,1721.57 \n 1955.1,1726.48 1964.46,1729.92 1974.23,1731.8 1984.16,1732.06 1994.01,1730.71 2003.53,1727.78 2012.48,1723.33 2020.63,1717.48 2027.77,1710.37 2033.73,1702.18 \n 2038.34,1693.12 2041.49,1683.43 2043.1,1673.34 2043.12,1663.11 2041.55,1653.01 2038.43,1643.31 2033.83,1634.24 2027.89,1626.05 2020.74,1618.95 2012.57,1613.12 \n 2003.6,1608.72 1994.05,1605.87 1984.19,1604.65 1974.26,1605.08 1964.53,1607.16 1955.25,1610.83 1946.68,1616.01 1939.04,1622.55 1932.54,1630.27 1927.34,1638.99 \n 1923.6,1648.46 1921.41,1658.44 1920.84,1668.64 1921.89,1678.81 1924.55,1688.66 1928.75,1697.93 1934.36,1706.36 1941.25,1713.73 1949.23,1719.83 1958.07,1724.5 \n 1967.54,1727.6 1977.38,1729.05 1987.31,1728.8 1997.07,1726.86 2006.38,1723.29 2014.98,1718.17 2022.64,1711.65 2029.13,1703.92 2034.29,1695.18 2037.95,1685.67 \n 2040.02,1675.67 2040.44,1665.46 2039.18,1655.32 2036.3,1645.53 2031.85,1636.39 2025.98,1628.14 2018.85,1621.03 2010.65,1615.25 2001.62,1610.98 1992.02,1608.34 \n 1982.13,1607.4 1972.22,1608.2 1962.59,1610.72 1953.51,1614.88 1945.25,1620.56 1938.05,1627.61 1932.12,1635.81 1927.64,1644.93 1924.73,1654.71 1923.49,1664.85 \n 1923.96,1675.07 1926.11,1685.05 1929.9,1694.5 1935.2,1703.14 1941.87,1710.72 1949.71,1717.01 1958.48,1721.81 1967.92,1724.99 1977.76,1726.44 1987.69,1726.12 \n 1997.42,1724.05 2006.65,1720.26 2015.1,1714.89 2022.52,1708.1 2028.68,1700.08 2033.4,1691.08 2036.51,1681.37 2037.93,1671.25 2037.62,1661.03 2035.58,1651.03 \n 2031.86,1641.55 2026.59,1632.88 2019.93,1625.3 2012.08,1619.04 2003.27,1614.29 1993.8,1611.22 1983.95,1609.9 1974.02,1610.4 1964.34,1612.69 1955.2,1616.7 \n 1946.9,1622.31 1939.69,1629.35 1933.81,1637.59 1929.44,1646.77 1926.73,1656.61 1925.77,1666.78 1926.57,1676.97 1929.13,1686.84 1933.37,1696.09 1939.13,1704.42 \n 1946.26,1711.54 1954.51,1717.24 1963.61,1721.33 1973.28,1723.66 1983.21,1724.18 1993.06,1722.85 2002.51,1719.71 2011.27,1714.87 2019.03,1708.49 2025.54,1700.77 \n 2030.59,1691.97 2034.01,1682.37 2035.68,1672.3 2035.55,1662.08 2033.61,1652.05 2029.94,1642.55 2024.64,1633.9 2017.91,1626.39 2009.95,1620.27 2001.05,1615.74 \n 1991.49,1612.96 1981.59,1612.04 1971.7,1612.99 1962.15,1615.79 1953.25,1620.35 1945.33,1626.51 1938.64,1634.07 1933.42,1642.76 1929.84,1652.3 1928.04,1662.35 \n 1928.07,1672.57 1929.94,1682.61 1933.58,1692.12 1938.87,1700.77 1945.63,1708.26 1953.63,1714.32 1962.58,1718.76 1972.17,1721.4 1982.08,1722.15 1991.95,1720.99 \n 2001.44,1717.96 2010.21,1713.15 2017.94,1706.75 2024.38,1698.96 2029.28,1690.07 2032.48,1680.39 2033.85,1670.27 2033.34,1660.06 2030.98,1650.14 2026.84,1640.84 \n 2021.07,1632.52 2013.88,1625.47 2005.52,1619.95 1996.3,1616.15 1986.54,1614.22 1976.61,1614.23 1966.86,1616.18 1957.64,1620 1949.3,1625.56 1942.15,1632.64 \n 1936.43,1641 1932.37,1650.33 1930.12,1660.28 1929.76,1670.5 1931.3,1680.59 1934.7,1690.2 1939.83,1698.95 1946.49,1706.53 1954.44,1712.65 1963.39,1717.08 \n 1973.01,1719.66 1982.92,1720.28 1992.77,1718.92 2002.17,1715.63 2010.78,1710.53 2018.26,1703.81 2024.34,1695.73 2028.78,1686.59 2031.41,1676.73 2032.13,1666.54 \n 2030.91,1656.4 2027.79,1646.69 2022.89,1637.8 2016.4,1630.07 2008.56,1623.78 1999.69,1619.2 1990.11,1616.49 1980.2,1615.76 1970.35,1617.05 1960.93,1620.3 \n 1952.32,1625.39 1944.85,1632.13 1938.82,1640.25 1934.46,1649.43 1931.94,1659.31 1931.38,1669.52 1932.78,1679.63 1936.1,1689.26 1941.21,1698.03 1947.91,1705.57 \n 1955.93,1711.6 1964.96,1715.87 1974.62,1718.21 1984.55,1718.52 1994.34,1716.78 2003.6,1713.08 2011.95,1707.55 2019.05,1700.41 2024.63,1691.95 2028.45,1682.52 \n 2030.35,1672.49 2030.26,1662.27 2028.17,1652.28 2024.18,1642.93 2018.44,1634.59 2011.18,1627.61 2002.71,1622.28 1993.36,1618.82 1983.53,1617.38 1973.62,1618.01 \n 1964.03,1620.69 1955.18,1625.31 1947.42,1631.69 1941.07,1639.55 1936.42,1648.57 1933.64,1658.39 1932.86,1668.57 1934.12,1678.71 1937.36,1688.37 1942.45,1697.14 \n 1949.17,1704.66 1957.25,1710.61 1966.33,1714.73 1976.05,1716.84 1985.98,1716.86 1995.7,1714.78 2004.8,1710.69 2012.89,1704.76 2019.62,1697.25 2024.7,1688.47 \n 2027.91,1678.8 2029.1,1668.65 2028.24,1658.48 2025.34,1648.7 2020.55,1639.76 2014.05,1632.03 2006.14,1625.86 1997.15,1621.51 1987.48,1619.18 1977.55,1618.97 \n 1967.8,1620.89 1958.65,1624.86 1950.51,1630.71 1943.73,1638.18 1938.62,1646.94 1935.41,1656.61 1934.24,1666.76 1935.16,1676.93 1938.13,1686.68 1943.03,1695.56 \n 1949.63,1703.2 1957.65,1709.23 1966.72,1713.38 1976.44,1715.48 1986.36,1715.42 1996.06,1713.2 2005.08,1708.93 2013.01,1702.79 2019.5,1695.06 2024.25,1686.08 \n 2027.05,1676.28 2027.75,1666.09 2026.34,1655.98 2022.86,1646.41 2017.49,1637.82 2010.45,1630.6 2002.09,1625.1 1992.77,1621.56 1982.94,1620.15 1973.04,1620.94 \n 1963.53,1623.89 1954.86,1628.87 1947.43,1635.65 1941.59,1643.91 1937.61,1653.27 1935.68,1663.29 1935.89,1673.5 1938.24,1683.43 1942.61,1692.6 1948.8,1700.59 \n 1956.53,1707.01 1965.42,1711.56 1975.05,1714.02 1984.98,1714.27 1994.72,1712.31 2003.82,1708.21 2011.83,1702.18 2018.38,1694.5 2023.14,1685.54 2025.9,1675.72 \n 2026.51,1665.53 2024.94,1655.44 2021.26,1645.95 2015.66,1637.52 2008.41,1630.55 1999.84,1625.38 1990.38,1622.27 1980.49,1621.38 1970.65,1622.74 1961.34,1626.29 \n 1953.02,1631.86 1946.1,1639.18 1940.91,1647.89 1937.72,1657.56 1936.68,1667.72 1937.85,1677.87 1941.17,1687.49 1946.48,1696.12 1953.52,1703.33 1961.93,1708.75 \n 1971.31,1712.11 1981.17,1713.25 1991.04,1712.11 2000.41,1708.73 2008.81,1703.29 2015.82,1696.06 2021.09,1687.4 2024.35,1677.76 2025.43,1667.6 2024.27,1657.46 \n 2020.94,1647.84 2015.59,1639.23 2008.51,1632.07 2000.04,1626.74 1990.63,1623.49 1980.75,1622.51 1970.91,1623.84 1961.61,1627.42 1953.33,1633.07 1946.51,1640.48 \n 1941.49,1649.3 1938.54,1659.05 1937.81,1669.23 1939.33,1679.33 1943.04,1688.8 1948.75,1697.16 1956.15,1703.97 1964.85,1708.87 1974.42,1711.61 1984.34,1712.03 \n 1994.09,1710.12 2003.16,1705.98 2011.07,1699.81 2017.41,1691.95 2021.83,1682.8 2024.09,1672.86 2024.09,1662.64 2021.81,1652.7 2017.38,1643.57 2011.03,1635.72 \n 2003.09,1629.58 1994,1625.49 1984.23,1623.66 1974.32,1624.19 1964.79,1627.06 1956.17,1632.11 1948.91,1639.08 1943.41,1647.58 1939.97,1657.16 1938.78,1667.3 \n 1939.91,1677.45 1943.29,1687.05 1948.75,1695.58 1955.98,1702.57 1964.6,1707.65 1974.12,1710.52 1984.03,1711.03 1993.79,1709.15 2002.85,1704.98 2010.72,1698.75 \n 2016.95,1690.8 2021.2,1681.57 2023.24,1671.58 2022.94,1661.37 2020.33,1651.52 2015.54,1642.57 2008.85,1635.03 2000.62,1629.32 1991.32,1625.76 1981.46,1624.56 \n 1971.61,1625.77 1962.31,1629.35 1954.09,1635.07 1947.42,1642.64 1942.68,1651.61 1940.14,1661.48 1939.93,1671.69 1942.09,1681.65 1946.48,1690.81 1952.85,1698.64 \n 1960.85,1704.68 1970.02,1708.59 1979.83,1710.15 1989.72,1709.26 1999.11,1705.97 2007.47,1700.47 2014.32,1693.08 2019.25,1684.21 2021.98,1674.39 2022.34,1664.19 \n 2020.32,1654.19 2016.03,1644.99 2009.72,1637.11 2001.76,1631.01 1992.6,1627.07 1982.8,1625.5 1972.91,1626.41 1963.53,1629.75 1955.21,1635.31 1948.43,1642.77 \n 1943.61,1651.69 1941.02,1661.55 1940.83,1671.76 1943.04,1681.71 1947.53,1690.81 1954.03,1698.53 1962.16,1704.39 1971.43,1708.05 1981.28,1709.28 1991.12,1708.01 \n 2000.37,1704.31 2008.47,1698.4 2014.92,1690.65 2019.34,1681.51 2021.46,1671.53 2021.15,1661.33 2018.41,1651.51 2013.43,1642.68 2006.5,1635.37 1998.04,1630.04 \n 1988.57,1627 1978.66,1626.45 1968.92,1628.42 1959.95,1632.79 1952.31,1639.3 1946.46,1647.55 1942.76,1657.02 1941.45,1667.14 1942.6,1677.28 1946.16,1686.81 \n 1951.89,1695.15 1959.45,1701.76 1968.37,1706.24 1978.08,1708.31 1988,1707.83 1997.48,1704.84 2005.95,1699.52 2012.87,1692.2 2017.79,1683.34 2020.42,1673.49 \n 2020.59,1663.29 2018.27,1653.36 2013.62,1644.34 2006.93,1636.8 1998.62,1631.23 1989.22,1627.96 1979.32,1627.23 1969.56,1629.07 1960.56,1633.36 1952.89,1639.84 \n 1947.06,1648.1 1943.42,1657.6 1942.23,1667.73 1943.55,1677.85 1947.31,1687.3 1953.25,1695.47 1961.01,1701.83 1970.08,1705.97 1979.87,1707.62 1989.75,1706.67 \n 1999.08,1703.17 2007.23,1697.36 2013.7,1689.62 2018.04,1680.44 2019.97,1670.43 2019.38,1660.24 2016.28,1650.54 2010.89,1641.97 2003.55,1635.1 1994.76,1630.38 \n 1985.08,1628.11 1975.17,1628.47 1965.67,1631.41 1957.22,1636.76 1950.38,1644.15 1945.6,1653.1 1943.21,1663.01 1943.37,1673.21 1946.07,1683.03 1951.13,1691.82 \n 1958.21,1698.96 1966.84,1704 1976.44,1706.58 1986.36,1706.54 1995.93,1703.86 2004.51,1698.74 2011.51,1691.51 2016.46,1682.66 2019.02,1672.8 2019.02,1662.59 \n 2016.45,1652.74 2011.48,1643.9 2004.46,1636.69 1995.86,1631.6 1986.27,1628.98 1976.35,1629.02 1966.78,1631.71 1958.22,1636.86 1951.26,1644.13 1946.38,1653.02 \n 1943.93,1662.91 1944.06,1673.12 1946.78,1682.93 1951.9,1691.67 1959.06,1698.73 1967.77,1703.62 1977.42,1705.99 1987.34,1705.67 1996.82,1702.69 2005.22,1697.25 \n 2011.93,1689.73 2016.48,1680.67 2018.56,1670.69 2018,1660.5 2014.86,1650.82 2009.35,1642.33 2001.86,1635.64 1992.92,1631.22 1983.16,1629.4 1973.28,1630.29 \n 1963.98,1633.84 1955.93,1639.79 1949.69,1647.73 1945.73,1657.09 1944.32,1667.19 1945.58,1677.31 1949.4,1686.73 1955.51,1694.76 1963.49,1700.83 1972.75,1704.5 \n 1982.62,1705.49 1992.39,1703.73 2001.35,1699.36 2008.85,1692.68 2014.34,1684.19 2017.43,1674.49 2017.87,1664.29 2015.65,1654.35 2010.92,1645.38 2004.02,1638.05 \n 1995.46,1632.89 1985.87,1630.29 1975.96,1630.43 1966.44,1633.32 1958.04,1638.73 1951.36,1646.28 1946.91,1655.39 1945.01,1665.41 1945.81,1675.58 1949.25,1685.15 \n 1955.08,1693.41 1962.86,1699.73 1972.02,1703.65 1981.86,1704.87 1991.66,1703.3 2000.68,1699.05 2008.24,1692.44 2013.76,1683.97 2016.84,1674.27 2017.24,1664.08 \n 2014.93,1654.15 2010.07,1645.26 2003.04,1638.06 1994.37,1633.11 1984.71,1630.78 1974.81,1631.26 1965.4,1634.51 1957.23,1640.28 1950.9,1648.13 1946.9,1657.47 \n 1945.55,1667.58 1946.94,1677.68 1950.98,1687 1957.35,1694.82 1965.56,1700.53 1974.99,1703.7 1984.9,1704.08 1994.53,1701.63 2003.12,1696.54 2010.02,1689.21 \n 2014.68,1680.21 2016.75,1670.23 2016.05,1660.05 2012.63,1650.47 2006.78,1642.24 1998.94,1636 1989.72,1632.24 1979.85,1631.26 1970.1,1633.14 1961.25,1637.73 \n 1953.98,1644.67 1948.88,1653.42 1946.35,1663.29 1946.6,1673.49 1949.6,1683.21 1955.11,1691.69 1962.71,1698.25 1971.78,1702.36 1981.61,1703.69 1991.41,1702.14 \n 2000.4,1697.83 2007.85,1691.11 2013.18,1682.5 2015.94,1672.7 2015.91,1662.5 2013.1,1652.72 2007.73,1644.14 2000.23,1637.46 1991.21,1633.23 1981.4,1631.79 \n 1971.59,1633.24 1962.57,1637.49 1955.09,1644.19 1949.75,1652.78 1946.99,1662.58 1947.04,1672.78 1949.89,1682.55 1955.31,1691.09 1962.87,1697.69 1971.93,1701.83 \n 1981.76,1703.14 1991.55,1701.52 2000.49,1697.11 2007.84,1690.27 2013,1681.56 2015.53,1671.69 2015.22,1661.5 2012.1,1651.82 2006.42,1643.45 1998.66,1637.11 \n 1989.46,1633.31 1979.58,1632.37 1969.87,1634.39 1961.11,1639.18 1954.06,1646.35 1949.31,1655.3 1947.24,1665.27 1948.05,1675.44 1951.67,1684.94 1957.78,1692.96 \n 1965.88,1698.84 1975.29,1702.07 1985.19,1702.38 1994.77,1699.73 2003.19,1694.36 2009.75,1686.71 2013.89,1677.44 2015.24,1667.34 2013.69,1657.26 2009.38,1648.08 \n 2002.67,1640.58 1994.13,1635.4 1984.5,1632.98 1974.6,1633.55 1965.29,1637.05 1957.37,1643.19 1951.52,1651.42 1948.26,1661.05 1947.86,1671.24 1950.37,1681.11 \n 1955.56,1689.8 1962.99,1696.55 1972.01,1700.78 1981.84,1702.11 1991.62,1700.42 2000.49,1695.87 2007.68,1688.85 2012.55,1679.97 2014.68,1670.01 2013.88,1659.85 \n 2010.21,1650.37 2004.01,1642.42 1995.8,1636.7 1986.33,1633.71 1976.41,1633.72 1966.94,1636.73 1958.76,1642.48 1952.58,1650.46 1948.97,1659.95 1948.24,1670.12 \n 1950.46,1680.06 1955.44,1688.88 1962.73,1695.79 1971.68,1700.17 1981.49,1701.63 1991.28,1700.03 2000.17,1695.52 2007.35,1688.5 2012.19,1679.6 2014.24,1669.62 \n 2013.31,1659.47 2009.49,1650.06 2003.13,1642.24 1994.78,1636.73 1985.23,1634.03 1975.32,1634.39 1965.97,1637.77 1958.03,1643.87 1952.22,1652.13 1949.08,1661.8 \n 1948.89,1671.99 1951.67,1681.78 1957.18,1690.25 1964.9,1696.64 1974.13,1700.35 1984.02,1701.05 1993.65,1698.65 2002.14,1693.39 2008.7,1685.74 2012.72,1676.42 \n 2013.82,1666.29 2011.9,1656.28 2007.15,1647.34 1999.99,1640.29 1991.1,1635.79 1981.3,1634.26 1971.51,1635.85 1962.65,1640.41 1955.55,1647.52 1950.86,1656.5 \n 1949.05,1666.52 1950.27,1676.64 1954.42,1685.9 1961.11,1693.43 1969.7,1698.51 1979.38,1700.66 1989.25,1699.69 1998.35,1695.67 2005.84,1688.99 2010.99,1680.28 \n 2013.31,1670.37 2012.59,1660.2 2008.88,1650.74 2002.54,1642.91 1994.18,1637.44 1984.6,1634.86 1974.7,1635.42 1965.45,1639.08 1957.74,1645.47 1952.3,1653.99 \n 1949.66,1663.82 1950.08,1674 1953.52,1683.56 1959.65,1691.57 1967.88,1697.25 1977.41,1700.05 1987.31,1699.7 1996.62,1696.22 2004.44,1689.96 2010,1681.52 \n 2012.76,1671.73 2012.44,1661.54 2009.08,1651.95 2003,1643.9 1994.8,1638.18 1985.28,1635.36 1975.38,1635.71 1966.07,1639.2 1958.27,1645.49 1952.76,1653.95 \n 1950.07,1663.77 1950.48,1673.95 1953.95,1683.5 1960.13,1691.46 1968.42,1697.05 1977.98,1699.71 1987.88,1699.16 1997.11,1695.47 2004.76,1688.99 2010.07,1680.38 \n 2012.49,1670.5 2011.79,1660.33 2008.03,1650.9 2001.59,1643.15 1993.12,1637.87 1983.47,1635.59 1973.6,1636.53 1964.53,1640.62 1957.16,1647.43 1952.24,1656.27 \n 1950.27,1666.26 1951.46,1676.38 1955.68,1685.6 1962.51,1692.98 1971.26,1697.78 1981.02,1699.49 1990.81,1697.94 1999.63,1693.29 2006.56,1686.01 2010.91,1676.85 \n 2012.22,1666.75 2010.35,1656.74 2005.5,1647.86 1998.16,1641.01 1989.09,1636.91 1979.23,1635.98 1969.59,1638.33 1961.17,1643.7 1954.85,1651.54 1951.28,1661.05 \n 1950.83,1671.23 1953.56,1681.03 1959.18,1689.42 1967.1,1695.53 1976.51,1698.72 1986.42,1698.66 1995.78,1695.34 2003.63,1689.12 2009.13,1680.65 2011.71,1670.81 \n 2011.09,1660.64 2007.34,1651.21 2000.85,1643.51 1992.3,1638.36 1982.6,1636.31 1972.77,1637.56 1963.86,1642 1956.8,1649.15 1952.35,1658.26 1950.99,1668.35 \n 1952.86,1678.36 1957.76,1687.21 1965.18,1693.97 1974.32,1697.9 1984.2,1698.59 1993.76,1695.96 2001.99,1690.29 2008,1682.19 2011.13,1672.52 2011.06,1662.33 \n 2007.78,1652.72 2001.65,1644.71 1993.33,1639.19 1983.72,1636.74 1973.86,1637.63 1964.81,1641.77 1957.56,1648.71 1952.9,1657.7 1951.33,1667.76 1953.04,1677.8 \n 1957.82,1686.72 1965.18,1693.54 1974.29,1697.53 1984.17,1698.24 1993.73,1695.59 2001.93,1689.87 2007.86,1681.71 2010.86,1672 2010.62,1661.82 2007.15,1652.27 \n 2000.83,1644.43 1992.36,1639.15 1982.68,1637.02 1972.85,1638.28 1963.97,1642.79 1957.02,1650.05 1952.78,1659.26 1951.72,1669.39 1953.96,1679.31 1959.25,1687.92 \n 1967,1694.26 1976.36,1697.61 1986.26,1697.6 1995.6,1694.23 2003.34,1687.87 2008.6,1679.24 2010.79,1669.31 2009.66,1659.19 2005.34,1650.02 1998.31,1642.84 \n 1989.37,1638.47 1979.53,1637.39 1969.89,1639.74 1961.56,1645.24 1955.48,1653.28 1952.34,1662.94 1952.5,1673.12 1955.95,1682.67 1962.29,1690.5 1970.8,1695.7 \n 1980.51,1697.69 1990.31,1696.24 1999.07,1691.5 2005.79,1684.02 2009.71,1674.66 2010.35,1664.5 2007.66,1654.7 2001.93,1646.39 1993.83,1640.53 1984.29,1637.81 \n 1974.41,1638.53 1965.35,1642.62 1958.14,1649.6 1953.62,1658.66 1952.33,1668.76 1954.42,1678.72 1959.64,1687.38 1967.38,1693.72 1976.75,1697.01 1986.65,1696.87 \n 1995.92,1693.29 2003.47,1686.71 2008.43,1677.9 2010.2,1667.88 2008.58,1657.83 2003.75,1648.94 1996.28,1642.25 1987.06,1638.56 1977.16,1638.29 1967.76,1641.5 \n 1959.98,1647.79 1954.73,1656.42 1952.64,1666.38 1953.96,1676.47 1958.53,1685.51 1965.81,1692.4 1974.94,1696.34 1984.83,1696.84 1994.29,1693.86 2002.2,1687.73 \n 2007.6,1679.2 2009.86,1669.28 2008.69,1659.17 2004.23,1650.07 1997.02,1643.09 1987.93,1639.07 1978.05,1638.48 1968.57,1641.41 1960.64,1647.5 1955.21,1656.02 \n 1952.95,1665.94 1954.14,1676.05 1958.62,1685.13 1965.86,1692.07 1974.97,1696.04 1984.86,1696.54 1994.31,1693.51 2002.17,1687.32 2007.48,1678.72 2009.59,1668.77 \n 2008.24,1658.69 2003.58,1649.7 1996.21,1642.91 1987,1639.16 1977.11,1638.91 1967.74,1642.2 1960.05,1648.61 1955,1657.36 1953.2,1667.38 1954.88,1677.41 \n 1959.84,1686.23 1967.46,1692.73 1976.79,1696.1 1986.69,1695.94 1995.91,1692.24 2003.31,1685.49 2007.97,1676.5 2009.3,1666.41 2007.14,1656.48 2001.74,1647.94 \n 1993.8,1641.87 1984.3,1639.03 1974.43,1639.77 1965.43,1644.01 1958.44,1651.22 1954.34,1660.49 1953.64,1670.64 1956.44,1680.41 1962.38,1688.55 1970.71,1694.04 \n 1980.39,1696.18 1990.18,1694.69 1998.84,1689.76 2005.27,1682.03 2008.65,1672.46 2008.56,1662.28 2004.99,1652.78 1998.41,1645.18 1989.65,1640.45 1979.83,1639.19 \n 1970.21,1641.57 1962.02,1647.29 1956.31,1655.61 1953.82,1665.46 1954.87,1675.58 1959.33,1684.67 1966.62,1691.56 1975.8,1695.35 1985.7,1695.55 1995.02,1692.14 \n 2002.56,1685.55 2007.35,1676.65 2008.77,1666.57 2006.61,1656.63 2001.18,1648.13 1993.16,1642.16 1983.61,1639.51 1973.76,1640.53 1964.91,1645.08 1958.21,1652.57 \n 1954.54,1662.03 1954.39,1672.2 1957.76,1681.77 1964.23,1689.48 1972.94,1694.3 1982.75,1695.61 1992.37,1693.22 2000.53,1687.46 2006.15,1679.09 2008.49,1669.19 \n 2007.23,1659.1 2002.54,1650.13 1995.05,1643.49 1985.74,1640.05 1975.85,1640.27 1966.69,1644.13 1959.49,1651.11 1955.21,1660.28 1954.41,1670.43 1957.21,1680.19 \n 1963.23,1688.26 1971.68,1693.56 1981.41,1695.38 1991.13,1693.48 1999.52,1688.09 2005.47,1679.96 2008.16,1670.17 2007.24,1660.03 2002.82,1650.92 1995.51,1644.07 \n 1986.28,1640.41 1976.39,1640.42 1967.17,1644.12 1959.88,1651 1955.51,1660.12 1954.64,1670.26 1957.41,1680.03 1963.43,1688.11 1971.9,1693.38 1981.64,1695.13 \n 1991.34,1693.12 1999.66,1687.62 2005.47,1679.38 2007.97,1669.54 2006.81,1659.43 2002.15,1650.45 1994.64,1643.83 1985.3,1640.49 1975.41,1640.89 1966.35,1644.97 \n 1959.36,1652.17 1955.42,1661.5 1955.06,1671.67 1958.34,1681.27 1964.8,1688.97 1973.56,1693.7 1983.39,1694.8 1992.93,1692.12 2000.85,1686.02 2006.05,1677.36 \n 2007.79,1667.34 2005.84,1657.37 2000.46,1648.83 1992.4,1642.92 1982.8,1640.48 1973,1641.84 1964.38,1646.81 1958.14,1654.71 1955.17,1664.41 1955.88,1674.56 \n 1960.19,1683.72 1967.47,1690.6 1976.71,1694.23 1986.6,1694.09 1995.74,1690.19 2002.83,1683.1 2006.86,1673.81 2007.27,1663.64 2003.99,1654.05 1997.48,1646.38 \n 1988.68,1641.75 1978.84,1640.8 1969.35,1643.68 1961.59,1649.98 1956.65,1658.79 1955.25,1668.87 1957.59,1678.75 1963.34,1687.03 1971.67,1692.51 1981.38,1694.41 \n 1991.09,1692.45 1999.38,1686.9 2005.06,1678.58 2007.31,1668.67 2005.81,1658.61 2000.77,1649.87 1992.91,1643.69 1983.38,1640.98 1973.55,1642.15 1964.86,1647 \n 1958.58,1654.86 1955.61,1664.56 1956.39,1674.7 1960.81,1683.8 1968.23,1690.53 1977.55,1693.9 1987.43,1693.42 1996.41,1689.16 2003.17,1681.74 2006.72,1672.24 \n 2006.54,1662.07 2002.65,1652.72 1995.62,1645.57 1986.49,1641.66 1976.61,1641.59 1967.43,1645.36 1960.3,1652.41 1956.3,1661.71 1956,1671.88 1959.47,1681.41 \n 1966.18,1688.88 1975.13,1693.18 1985.01,1693.68 1994.33,1690.29 2001.71,1683.52 2006.04,1674.38 2006.68,1664.23 2003.53,1654.59 1997.06,1646.9 1988.24,1642.32 \n 1978.38,1641.53 1968.97,1644.66 1961.43,1651.23 1956.88,1660.26 1956.02,1670.39 1958.98,1680.1 1965.3,1687.91 1974.05,1692.65 1983.89,1693.61 1993.34,1690.62 \n 2000.96,1684.15 2005.61,1675.17 2006.56,1665.05 2003.69,1655.32 1997.41,1647.46 1988.69,1642.68 1978.85,1641.69 1969.39,1644.66 1961.77,1651.13 1957.14,1660.12 \n 1956.21,1670.25 1959.14,1679.96 1965.46,1687.77 1974.22,1692.49 1984.07,1693.37 1993.49,1690.29 2001.03,1683.72 2005.54,1674.67 2006.3,1664.53 2003.21,1654.87 \n 1996.74,1647.19 1987.89,1642.66 1978.03,1642 1968.68,1645.31 1961.31,1652.08 1957.04,1661.25 1956.56,1671.41 1959.94,1680.97 1966.64,1688.44 1975.64,1692.65 \n 1985.51,1692.96 1994.73,1689.29 2001.85,1682.24 2005.75,1672.9 2005.83,1662.73 2002.05,1653.33 1995.03,1646.18 1985.86,1642.4 1975.98,1642.58 1966.94,1646.71 \n 1960.19,1654.13 1956.77,1663.67 1957.23,1673.83 1961.51,1682.99 1968.92,1689.72 1978.3,1692.93 1988.15,1692.13 1996.91,1687.43 2003.2,1679.58 2006,1669.83 \n 2004.87,1659.73 1999.99,1650.89 1992.14,1644.72 1982.56,1642.21 1972.8,1643.75 1964.4,1649.12 1958.72,1657.44 1956.67,1667.38 1958.58,1677.36 1964.14,1685.76 \n 1972.47,1691.24 1982.21,1692.91 1991.81,1690.5 1999.72,1684.4 2004.65,1675.59 2005.81,1665.49 2003.01,1655.74 1996.71,1647.92 1987.91,1643.29 1978.05,1642.6 \n 1968.73,1645.98 1961.46,1652.87 1957.43,1662.15 1957.3,1672.31 1961.08,1681.7 1968.16,1688.79 1977.4,1692.41 1987.27,1691.97 1996.16,1687.55 2002.62,1679.86 \n 2005.59,1670.16 2004.57,1660.05 1999.74,1651.18 1991.88,1645.02 1982.29,1642.58 1972.55,1644.26 1964.26,1649.79 1958.79,1658.26 1957.05,1668.26 1959.33,1678.15 \n 1965.25,1686.29 1973.83,1691.33 1983.65,1692.43 1993.08,1689.4 2000.56,1682.75 2004.83,1673.59 2005.19,1663.43 2001.57,1653.97 1994.57,1646.8 1985.37,1643.09 \n 1975.5,1643.49 1966.61,1647.92 1960.19,1655.64 1957.32,1665.37 1958.48,1675.46 1963.47,1684.23 1971.47,1690.2 1981.12,1692.36 1990.8,1690.35 1998.88,1684.51 \n 2004,1675.81 2005.28,1665.73 2002.52,1655.98 1996.17,1648.19 1987.31,1643.7 1977.44,1643.26 1968.24,1646.95 1961.26,1654.15 1957.71,1663.63 1958.17,1673.78 \n 1962.59,1682.88 1970.19,1689.36 1979.7,1692.12 1989.48,1690.7 1997.86,1685.32 2003.41,1676.91 2005.17,1666.92 2002.84,1657.04 1996.83,1648.98 1988.15,1644.13 \n 1978.31,1643.31 1968.99,1646.68 1961.8,1653.65 1957.99,1663.02 1958.2,1673.18 1962.42,1682.37 1969.9,1689 1979.36,1691.92 1989.16,1690.63 1997.59,1685.34 \n 2003.2,1676.98 2005,1666.99 2002.69,1657.11 1996.66,1649.06 1987.96,1644.25 1978.11,1643.51 1968.83,1646.98 1961.74,1654.05 1958.08,1663.49 1958.49,1673.64 \n 1962.91,1682.73 1970.55,1689.16 1980.09,1691.8 1989.84,1690.19 1998.09,1684.6 2003.38,1676.02 2004.79,1665.97 2002.06,1656.2 1995.67,1648.46 1986.75,1644.1 \n 1976.88,1643.9 1967.8,1647.89 1961.12,1655.38 1958.04,1665.03 1959.09,1675.13 1964.09,1683.89 1972.16,1689.75 1981.86,1691.66 1991.46,1689.28 1999.25,1683.03 \n 2003.83,1674.04 2004.4,1663.9 2000.83,1654.42 1993.78,1647.31 1984.5,1643.84 1974.66,1644.63 1966.02,1649.55 1960.14,1657.7 1958.07,1667.64 1960.19,1677.56 \n 1966.12,1685.68 1974.79,1690.54 1984.64,1691.25 1993.88,1687.69 2000.86,1680.5 2004.29,1670.97 2003.56,1660.84 1998.8,1651.94 1990.88,1645.89 1981.22,1643.78 \n 1971.59,1646.01 1963.73,1652.16 1959.09,1661.13 1958.51,1671.27 1962.09,1680.73 1969.18,1687.79 1978.5,1691.16 1988.33,1690.21 1996.87,1685.12 2002.56,1676.82 \n 2004.35,1666.83 2001.91,1656.99 1995.69,1649.1 1986.83,1644.62 1976.96,1644.38 1967.91,1648.42 1961.33,1655.99 1958.45,1665.71 1959.81,1675.77 1965.14,1684.31 \n 1973.47,1689.76 1983.26,1691.09 1992.68,1688.07 1999.99,1681.24 2003.83,1671.89 2003.48,1661.74 1999.01,1652.69 1991.24,1646.42 1981.63,1644.1 1971.97,1646.18 \n 1964.06,1652.26 1959.39,1661.2 1958.83,1671.34 1962.48,1680.77 1969.67,1687.73 1979.05,1690.9 1988.85,1689.7 1997.23,1684.34 2002.61,1675.83 2003.99,1665.77 \n 2001.09,1656.07 1994.46,1648.54 1985.36,1644.62 1975.5,1645.05 1966.75,1649.74 1960.76,1657.82 1958.68,1667.74 1960.9,1677.64 1967,1685.62 1975.82,1690.17 \n 1985.68,1690.43 1994.72,1686.34 2001.19,1678.68 2003.88,1668.9 2002.25,1658.89 1996.63,1650.54 1988.09,1645.47 1978.25,1644.63 1969.01,1648.19 1962.14,1655.48 \n 1958.96,1665.09 1960.07,1675.18 1965.28,1683.8 1973.57,1689.3 1983.35,1690.62 1992.74,1687.5 1999.93,1680.54 2003.51,1671.08 2002.81,1660.95 1997.95,1652.12 \n 1989.87,1646.29 1980.15,1644.59 1970.65,1647.36 1963.24,1654.06 1959.34,1663.38 1959.73,1673.53 1964.32,1682.51 1972.23,1688.59 1981.91,1690.56 1991.47,1688.06 \n 1999.04,1681.55 2003.15,1672.32 2002.99,1662.17 1998.58,1653.09 1990.79,1646.86 1981.15,1644.7 1971.55,1647.04 1963.88,1653.43 1959.65,1662.6 1959.69,1672.75 \n 1964,1681.88 1971.72,1688.19 1981.35,1690.43 1990.96,1688.15 1998.67,1681.82 2002.94,1672.67 2002.92,1662.52 1998.63,1653.38 1990.9,1647.07 1981.27,1644.84 \n 1971.66,1647.14 1963.99,1653.51 1959.77,1662.69 1959.85,1672.84 1964.22,1681.94 1972.01,1688.17 1981.65,1690.28 1991.23,1687.86 1998.82,1681.38 2002.91,1672.14 \n 2002.66,1661.99 1998.13,1652.98 1990.22,1646.91 1980.53,1645 1971.01,1647.64 1963.57,1654.31 1959.72,1663.65 1960.23,1673.79 1965,1682.67 1973.08,1688.5 \n 1982.82,1690.09 1992.25,1687.14 1999.47,1680.22 2003,1670.74 2002.13,1660.63 1997.04,1651.95 1988.75,1646.45 1978.95,1645.26 1969.65,1648.62 1962.73,1655.85 \n 1959.61,1665.48 1960.94,1675.54 1966.43,1683.96 1974.97,1689.04 1984.81,1689.72 1993.93,1685.87 2000.46,1678.27 2003.07,1668.48 2001.2,1658.51 1995.25,1650.42 \n 1986.43,1645.87 1976.57,1645.8 1967.7,1650.23 1961.64,1658.24 1959.66,1668.18 1962.16,1677.99 1968.62,1685.65 1977.72,1689.57 1987.56,1688.93 1996.11,1683.86 \n 2001.58,1675.42 2002.84,1665.36 1999.63,1655.77 1992.61,1648.64 1983.24,1645.47 1973.48,1646.92 1965.36,1652.68 1960.59,1661.55 1960.15,1671.69 1964.15,1680.96 \n 1971.74,1687.44 1981.34,1689.75 1990.94,1687.41 1998.51,1680.91 2002.47,1671.62 2001.98,1661.49 1997.15,1652.65 1988.98,1646.96 1979.21,1645.63 1969.89,1648.93 \n 1962.99,1656.18 1959.97,1665.84 1961.47,1675.86 1967.18,1684.13 1975.89,1688.9 1985.74,1689.14 1994.66,1684.81 2000.74,1676.82 2002.69,1666.88 2000.09,1657.09 \n 1993.5,1649.55 1984.32,1645.86 1974.5,1646.81 1966.15,1652.2 1961.05,1660.88 1960.29,1670.99 1964.03,1680.37 1971.48,1687.01 1981.04,1689.49 1990.66,1687.26 \n 1998.27,1680.82 2002.24,1671.53 2001.7,1661.4 1996.78,1652.61 1988.53,1647.06 1978.73,1645.94 1969.5,1649.5 1962.83,1656.96 1960.16,1666.73 1962.08,1676.68 \n 1968.17,1684.65 1977.11,1688.93 1986.96,1688.58 1995.58,1683.67 2001.11,1675.27 2002.33,1665.2 1998.97,1655.67 1991.78,1648.73 1982.31,1645.92 1972.63,1647.84 \n 1964.86,1654.07 1960.69,1663.26 1961.04,1673.4 1965.83,1682.26 1974.02,1687.9 1983.81,1689.09 1993.05,1685.56 1999.71,1678.09 2002.33,1668.31 2000.32,1658.38 \n 1994.13,1650.49 1985.13,1646.37 1975.29,1646.94 1966.79,1652.07 1961.5,1660.63 1960.61,1670.73 1964.3,1680.13 1971.77,1686.75 1981.35,1689.12 1990.92,1686.72 \n 1998.36,1680.07 2002.01,1670.65 2001.06,1660.56 1995.72,1652.04 1987.17,1646.99 1977.33,1646.54 1968.38,1650.79 1962.33,1658.8 1960.53,1668.77 1963.39,1678.47 \n 1970.26,1685.73 1979.62,1688.93 1989.35,1687.34 1997.28,1681.32 2001.62,1672.22 2001.4,1662.09 1996.66,1653.2 1988.48,1647.55 1978.68,1646.43 1969.49,1650.09 \n 1962.98,1657.69 1960.62,1667.54 1962.94,1677.39 1969.42,1685.02 1978.6,1688.7 1988.4,1687.6 1996.58,1681.96 2001.31,1673.06 2001.48,1662.93 1997.08,1653.86 \n 1989.09,1647.92 1979.34,1646.47 1970.05,1649.84 1963.34,1657.26 1960.73,1667.03 1962.84,1676.94 1969.18,1684.7 1978.29,1688.54 1988.1,1687.59 1996.35,1682.05 \n 2001.15,1673.2 2001.4,1663.06 1997.03,1653.98 1989.06,1648.03 1979.31,1646.59 1970.03,1649.99 1963.36,1657.45 1960.84,1667.25 1963.06,1677.13 1969.5,1684.8 \n 1978.67,1688.48 1988.46,1687.33 1996.59,1681.61 2001.18,1672.64 2001.16,1662.51 1996.54,1653.56 1988.38,1647.88 1978.58,1646.79 1969.44,1650.55 1963.06,1658.27 \n 1960.95,1668.17 1963.6,1677.93 1970.39,1685.28 1979.74,1688.48 1989.45,1686.8 1997.26,1680.63 2001.33,1671.4 2000.71,1661.29 1995.55,1652.66 1987.04,1647.54 \n 1977.2,1647.15 1968.34,1651.56 1962.53,1659.75 1961.17,1669.79 1964.55,1679.3 1971.9,1686.05 1981.47,1688.44 1991.01,1685.9 1998.25,1679.03 2001.48,1669.46 \n 1999.94,1659.45 1993.98,1651.38 1985.03,1647.16 1975.2,1647.8 1966.83,1653.15 1961.92,1661.93 1961.63,1672.06 1966.04,1681.12 1974.09,1686.95 1983.87,1688.16 \n 1993.04,1684.46 1999.4,1676.73 2001.44,1666.82 1998.66,1657.1 1991.74,1649.9 1982.31,1646.95 1972.66,1648.95 1965.1,1655.44 1961.44,1664.85 1962.57,1674.91 \n 1968.21,1683.21 1977.02,1687.75 1986.86,1687.42 1995.36,1682.31 2000.47,1673.65 2000.96,1663.53 1996.69,1654.4 1988.71,1648.47 1978.94,1647.18 1969.76,1650.84 \n 1963.39,1658.56 1961.38,1668.48 1964.22,1678.18 1971.22,1685.3 1980.68,1688.11 1990.3,1685.92 1997.72,1679.27 2001.14,1669.77 1999.72,1659.74 1993.81,1651.64 \n 1984.85,1647.45 1975.03,1648.19 1966.76,1653.68 1962.07,1662.59 1962.1,1672.72 1966.86,1681.59 1975.17,1687.01 1984.99,1687.66 1993.91,1683.38 1999.73,1675.2 \n 2001.01,1665.16 1997.43,1655.72 1989.88,1649.22 1980.23,1647.27 1970.84,1650.34 1964.06,1657.68 1961.56,1667.48 1963.96,1677.3 1970.67,1684.71 1980.02,1687.87 \n 1989.7,1686 1997.29,1679.55 2000.9,1670.13 1999.64,1660.08 1993.82,1651.92 1984.89,1647.66 1975.07,1648.39 1966.82,1653.91 1962.2,1662.86 1962.37,1672.98 \n 1967.29,1681.76 1975.72,1686.98 1985.56,1687.34 1994.33,1682.75 1999.83,1674.35 2000.69,1664.26 1996.67,1655.02 1988.79,1648.94 1979.04,1647.57 1969.87,1651.25 \n 1963.6,1659.05 1961.8,1669.01 1964.95,1678.61 1972.23,1685.41 1981.82,1687.71 1991.27,1684.91 1998.21,1677.73 2000.86,1667.98 1998.55,1658.13 1991.87,1650.7 \n 1982.51,1647.57 1972.86,1649.53 1965.37,1656.1 1961.95,1665.59 1963.47,1675.59 1969.56,1683.55 1978.65,1687.43 1988.42,1686.24 1996.37,1680.28 2000.48,1671.08 \n 1999.67,1660.99 1994.16,1652.6 1985.37,1648.06 1975.54,1648.54 1967.2,1653.92 1962.5,1662.82 1962.66,1672.94 1967.62,1681.68 1976.11,1686.79 1985.95,1686.95 \n 1994.59,1682.11 1999.81,1673.52 2000.24,1663.41 1995.78,1654.38 1987.58,1648.78 1977.77,1648.07 1968.88,1652.41 1963.23,1660.7 1962.29,1670.78 1966.29,1680.02 \n 1974.21,1686.03 1983.98,1687.24 1993.05,1683.33 1999.07,1675.32 2000.45,1665.3 1996.84,1655.88 1989.18,1649.53 1979.47,1647.91 1970.25,1651.44 1963.94,1659.2 \n 1962.19,1669.16 1965.46,1678.71 1972.9,1685.33 1982.55,1687.3 1991.87,1684.08 1998.42,1676.53 2000.47,1666.63 1997.47,1656.99 1990.22,1650.15 1980.63,1647.92 \n 1971.22,1650.89 1964.49,1658.27 1962.22,1668.12 1965.01,1677.83 1972.12,1684.82 1981.67,1687.24 1991.13,1684.46 1997.98,1677.19 2000.4,1667.39 1997.76,1657.64 \n 1990.74,1650.55 1981.22,1648 1971.74,1650.67 1964.81,1657.86 1962.3,1667.65 1964.88,1677.42 1971.85,1684.56 1981.35,1687.16 1990.85,1684.52 1997.79,1677.35 \n 2000.32,1667.57 1997.75,1657.81 1990.78,1650.67 1981.27,1648.09 1971.78,1650.75 1964.87,1657.95 1962.39,1667.74 1965.02,1677.49 1972.05,1684.57 1981.58,1687.07 \n 1991.04,1684.3 1997.87,1677.02 2000.23,1667.2 1997.47,1657.49 1990.34,1650.52 1980.77,1648.19 1971.36,1651.12 1964.66,1658.52 1962.49,1668.39 1965.45,1678.04 \n 1972.73,1684.85 1982.35,1686.95 1991.68,1683.78 1998.2,1676.2 2000.1,1666.27 1996.88,1656.72 1989.41,1650.14 1979.74,1648.33 1970.5,1651.81 1964.24,1659.61 \n 1962.66,1669.59 1966.21,1679.03 1973.91,1685.32 1983.64,1686.74 1992.74,1682.9 1998.69,1674.85 1999.86,1664.81 1995.93,1655.54 1987.97,1649.6 1978.19,1648.63 \n 1969.27,1652.9 1963.7,1661.23 1963,1671.32 1967.38,1680.37 1975.62,1685.89 1985.44,1686.33 1994.12,1681.58 1999.24,1672.95 1999.39,1662.84 1994.52,1654.05 \n 1985.98,1649.04 1976.15,1649.19 1967.77,1654.48 1963.17,1663.42 1963.64,1673.52 1969.05,1681.97 1977.89,1686.39 1987.69,1685.57 1995.7,1679.71 1999.69,1670.46 \n 1998.53,1660.42 1992.55,1652.39 1983.42,1648.64 1973.71,1650.21 1966.15,1656.67 1962.86,1666.2 1964.77,1676.12 1971.34,1683.63 1980.73,1686.63 1990.28,1684.26 \n 1997.31,1677.18 1999.82,1667.4 1997.1,1657.69 1989.92,1650.78 1980.32,1648.63 1971,1651.86 1964.62,1659.55 1962.98,1669.52 1966.56,1678.94 1974.32,1685.14 \n 1984.08,1686.34 1993.05,1682.21 1998.68,1673.93 1999.37,1663.84 1994.91,1654.83 1986.58,1649.47 1976.75,1649.28 1968.24,1654.33 1963.47,1663.18 1963.82,1673.28 \n 1969.18,1681.76 1978.03,1686.17 1987.81,1685.25 1995.74,1679.27 1999.52,1669.94 1998.07,1659.94 1991.81,1652.15 1982.54,1648.8 1972.92,1650.88 1965.73,1657.77 \n 1963.05,1667.49 1965.64,1677.24 1972.76,1684.21 1982.36,1686.37 1991.66,1683.1 1997.96,1675.34 1999.45,1665.35 1995.68,1656.01 1987.75,1650.04 1977.96,1649.17 \n 1969.16,1653.65 1963.89,1662.19 1963.71,1672.3 1968.65,1681.03 1977.29,1685.85 1987.1,1685.35 1995.23,1679.66 1999.29,1670.46 1998.11,1660.43 1992.02,1652.49 \n 1982.81,1648.98 1973.17,1650.93 1965.92,1657.75 1963.2,1667.47 1965.8,1677.21 1972.96,1684.13 1982.58,1686.19 1991.82,1682.78 1997.98,1674.91 1999.23,1664.88 \n 1995.21,1655.66 1987.1,1649.97 1977.28,1649.48 1968.67,1654.34 1963.8,1663.12 1964.12,1673.22 1969.53,1681.65 1978.44,1685.92 1988.2,1684.76 1995.91,1678.51 \n 1999.3,1669.02 1997.35,1659.12 1990.64,1651.74 1981.17,1649.08 1971.74,1651.93 1965.18,1659.45 1963.43,1669.39 1967.03,1678.8 1974.89,1684.86 1984.67,1685.76 \n 1993.45,1681.23 1998.6,1672.63 1998.59,1662.53 1993.4,1653.94 1984.6,1649.46 1974.82,1650.42 1967,1656.53 1963.49,1665.97 1965.34,1675.89 1972,1683.32 \n 1981.47,1686.01 1990.89,1683.16 1997.43,1675.62 1999.11,1665.66 1995.42,1656.29 1987.48,1650.35 1977.69,1649.62 1969,1654.33 1964.05,1663.05 1964.33,1673.15 \n 1969.77,1681.56 1978.71,1685.74 1988.45,1684.41 1996.02,1677.97 1999.12,1668.38 1996.81,1658.56 1989.79,1651.5 1980.2,1649.34 1970.96,1652.75 1964.88,1660.68 \n 1963.83,1670.73 1968.12,1679.81 1976.45,1685.16 1986.27,1685.14 1994.58,1679.74 1998.82,1670.63 1997.7,1660.6 1991.56,1652.71 1982.29,1649.4 1972.72,1651.68 \n 1965.81,1658.85 1963.67,1668.71 1966.97,1678.22 1974.69,1684.46 1984.46,1685.5 1993.25,1681.02 1998.37,1672.4 1998.22,1662.3 1992.85,1653.84 1983.93,1649.64 \n 1974.2,1651 1966.68,1657.5 1963.71,1667.13 1966.21,1676.9 1973.39,1683.78 1983.04,1685.63 1992.16,1681.88 1997.9,1673.69 1998.49,1663.61 1993.73,1654.77 \n 1985.11,1649.94 1975.32,1650.61 1967.4,1656.59 1963.84,1666 1965.75,1675.91 1972.53,1683.21 1982.06,1685.63 1991.36,1682.4 1997.51,1674.53 1998.59,1664.5 \n 1994.26,1655.44 1985.86,1650.2 1976.05,1650.43 1967.9,1656.06 1963.98,1665.32 1965.53,1675.29 1972.05,1682.84 1981.5,1685.58 1990.89,1682.64 1997.26,1674.96 \n 1998.6,1664.96 1994.48,1655.79 1986.2,1650.37 1976.38,1650.4 1968.14,1655.88 1964.08,1665.07 1965.5,1675.07 1971.94,1682.68 1981.36,1685.51 1990.77,1682.64 \n 1997.17,1674.99 1998.53,1664.99 1994.41,1655.82 1986.12,1650.42 1976.31,1650.49 1968.1,1656.03 1964.13,1665.26 1965.65,1675.23 1972.19,1682.76 1981.65,1685.44 \n 1991.01,1682.4 1997.26,1674.62 1998.4,1664.59 1994.07,1655.53 1985.65,1650.35 1975.84,1650.72 1967.8,1656.51 1964.12,1665.87 1965.99,1675.78 1972.79,1683.05 \n 1982.35,1685.34 1991.57,1681.9 1997.49,1673.85 1998.18,1663.78 1993.43,1654.95 1984.76,1650.21 1974.99,1651.11 1967.28,1657.35 1964.12,1666.91 1966.55,1676.69 \n 1973.78,1683.51 1983.46,1685.17 1992.44,1681.11 1997.81,1672.66 1997.81,1662.57 1992.45,1654.12 1983.46,1650.06 1973.79,1651.73 1966.58,1658.58 1964.19,1668.37 \n 1967.41,1677.9 1975.17,1684.07 1984.96,1684.84 1993.55,1679.96 1998.13,1671.04 1997.21,1661 1991.07,1653.12 1981.75,1650 1972.28,1652.66 1965.8,1660.24 \n 1964.43,1670.23 1968.63,1679.35 1977,1684.6 1986.81,1684.24 1994.8,1678.39 1998.34,1668.98 1996.27,1659.12 1989.26,1652.06 1979.63,1650.15 1970.57,1654.01 \n 1965.07,1662.37 1964.97,1672.46 1970.3,1680.94 1979.28,1684.98 1988.94,1683.25 1996.07,1676.33 1998.3,1666.5 1994.88,1657.04 1986.95,1651.1 1977.15,1650.66 \n 1968.75,1655.87 1964.55,1664.99 1965.96,1674.98 1972.5,1682.49 1981.99,1685.03 1991.26,1681.74 1997.2,1673.71 1997.83,1663.64 1992.92,1654.9 1984.14,1650.43 \n 1974.41,1651.71 1967.01,1658.34 1964.43,1668.07 1967.53,1677.64 1975.28,1683.83 1985.06,1684.55 1993.58,1679.56 1997.97,1670.54 1996.74,1660.53 1990.32,1652.91 \n 1980.86,1650.26 1971.56,1653.47 1965.58,1661.46 1964.93,1671.52 1969.83,1680.26 1978.64,1684.7 1988.35,1683.34 1995.68,1676.64 1998.13,1666.87 1994.88,1657.35 \n 1987.01,1651.33 1977.22,1650.84 1968.83,1656.06 1964.71,1665.21 1966.25,1675.17 1972.94,1682.54 1982.49,1684.81 1991.64,1681.2 1997.27,1672.94 1997.44,1662.86 \n 1992.1,1654.4 1983.07,1650.47 1973.45,1652.41 1966.54,1659.56 1964.71,1669.47 1968.58,1678.73 1976.84,1684.16 1986.64,1683.9 1994.61,1678.03 1998,1668.57 \n 1995.65,1658.78 1988.37,1652.03 1978.66,1650.66 1969.87,1655.14 1965.05,1663.92 1965.86,1673.97 1972.02,1681.81 1981.4,1684.73 1990.75,1681.72 1996.84,1673.82 \n 1997.54,1663.76 1992.62,1655.04 1983.79,1650.68 1974.1,1652.21 1966.93,1659.08 1964.77,1668.92 1968.38,1678.29 1976.5,1683.94 1986.3,1683.89 1994.37,1678.15 \n 1997.88,1668.74 1995.61,1658.93 1988.35,1652.16 1978.64,1650.78 1969.88,1655.3 1965.13,1664.12 1966.06,1674.15 1972.35,1681.89 1981.78,1684.61 1991.06,1681.35 \n 1996.91,1673.27 1997.29,1663.2 1992.05,1654.67 1983.05,1650.71 1973.44,1652.69 1966.62,1659.93 1964.99,1669.87 1969.13,1679 1977.57,1684.11 1987.35,1683.38 \n 1994.99,1677.07 1997.8,1667.42 1994.78,1657.83 1986.99,1651.7 1977.21,1651.22 1968.89,1656.54 1964.99,1665.79 1966.9,1675.68 1973.93,1682.69 1983.6,1684.33 \n 1992.46,1680.02 1997.35,1671.29 1996.53,1661.24 1990.3,1653.47 1980.87,1650.74 1971.61,1654.03 1965.82,1662.16 1965.59,1672.24 1970.98,1680.65 1980.08,1684.39 \n 1989.62,1682.12 1996.19,1674.64 1997.43,1664.64 1992.89,1655.71 1984.21,1651.05 1974.49,1652.34 1967.24,1659.11 1965.06,1668.94 1968.74,1678.28 1976.95,1683.77 \n 1986.74,1683.43 1994.58,1677.38 1997.63,1667.81 1994.8,1658.16 1987.1,1651.93 1977.32,1651.36 1969,1656.67 1965.14,1665.94 1967.15,1675.8 1974.3,1682.69 \n 1984,1684.1 1992.73,1679.52 1997.31,1670.62 1996.09,1660.62 1989.5,1653.16 1979.94,1650.96 1970.9,1654.82 1965.65,1663.33 1966.13,1673.4 1972.15,1681.34 \n 1981.52,1684.27 1990.82,1681.11 1996.65,1673.01 1996.87,1662.94 1991.41,1654.57 1982.26,1650.98 1972.77,1653.48 1966.43,1661.16 1965.55,1671.19 1970.47,1679.9 \n 1979.38,1684.09 1989,1682.22 1995.8,1674.97 1997.27,1665.01 1992.88,1656.01 1984.23,1651.28 1974.52,1652.57 1967.32,1659.41 1965.31,1669.26 1969.21,1678.5 \n 1977.59,1683.71 1987.35,1682.96 1994.88,1676.52 1997.4,1666.78 1993.95,1657.35 1985.83,1651.72 1976.05,1651.99 1968.22,1658.05 1965.28,1667.65 1968.3,1677.24 \n 1976.17,1683.23 1985.95,1683.4 1994.02,1677.69 1997.35,1668.22 1994.71,1658.52 1987.08,1652.21 1977.31,1651.65 1969.04,1657.05 1965.37,1666.38 1967.67,1676.17 \n 1975.09,1682.75 1984.84,1683.65 1993.27,1678.54 1997.22,1669.32 1995.21,1659.47 1987.99,1652.67 1978.27,1651.48 1969.71,1656.35 1965.52,1665.46 1967.28,1675.36 \n 1974.34,1682.34 1984.03,1683.76 1992.7,1679.09 1997.08,1670.08 1995.51,1660.14 1988.59,1653.02 1978.93,1651.42 1970.18,1655.93 1965.65,1664.86 1967.07,1674.82 \n 1973.89,1682.05 1983.53,1683.79 1992.33,1679.39 1996.96,1670.51 1995.64,1660.54 1988.89,1653.25 1979.27,1651.42 1970.43,1655.75 1965.75,1664.59 1967.01,1674.58 \n 1973.73,1681.9 1983.35,1683.76 1992.19,1679.45 1996.88,1670.62 1995.63,1660.63 1988.91,1653.32 1979.29,1651.47 1970.46,1655.8 1965.79,1664.65 1967.09,1674.63 \n 1973.85,1681.91 1983.48,1683.69 1992.28,1679.29 1996.87,1670.4 1995.49,1660.44 1988.65,1653.23 1979.01,1651.56 1970.26,1656.07 1965.78,1665.02 1967.31,1674.97 \n 1974.25,1682.06 1983.92,1683.57 1992.59,1678.91 1996.9,1669.87 1995.19,1659.96 1988.11,1653.01 1978.41,1651.72 1969.85,1656.58 1965.74,1665.71 1967.69,1675.58 \n 1974.93,1682.34 1984.66,1683.37 1993.1,1678.27 1996.95,1669.02 1994.72,1659.22 1987.28,1652.69 1977.52,1651.97 1969.25,1657.35 1965.7,1666.73 1968.26,1676.44 \n 1975.92,1682.7 1985.7,1683.05 1993.76,1677.36 1996.96,1667.84 1994.03,1658.24 1986.13,1652.31 1976.35,1652.38 1968.53,1658.42 1965.74,1668.06 1969.07,1677.52 \n 1977.23,1683.08 1986.99,1682.54 1994.53,1676.13 1996.86,1666.36 1993.08,1657.08 1984.66,1651.95 1974.94,1653 1967.74,1659.81 1965.91,1669.7 1970.18,1678.75 \n 1978.86,1683.39 1988.51,1681.78 1995.32,1674.55 1996.58,1664.58 1991.8,1655.8 1982.86,1651.71 1973.32,1653.93 1966.97,1661.57 1966.33,1671.61 1971.65,1680.05 \n 1980.82,1683.54 1990.2,1680.67 1996.04,1672.6 1996.01,1662.54 1990.13,1654.5 1980.74,1651.69 1971.59,1655.24 1966.34,1663.72 1967.08,1673.75 1973.52,1681.32 \n 1983.09,1683.39 1991.95,1679.14 1996.55,1670.26 1995.05,1660.32 1988.04,1653.31 1978.35,1652.03 1969.85,1657.01 1965.97,1666.24 1968.28,1676.02 1975.83,1682.4 \n 1985.6,1682.82 1993.65,1677.11 1996.72,1667.56 1993.59,1658.03 1985.51,1652.37 1975.75,1652.86 1968.24,1659.31 1966.04,1669.1 1970.02,1678.29 1978.58,1683.14 \n 1988.25,1681.7 1995.12,1674.54 1996.39,1664.57 1991.55,1655.83 1982.56,1651.87 1973.08,1654.31 1966.95,1662.15 1966.68,1672.2 1972.37,1680.38 1981.71,1683.35 \n 1990.89,1679.9 1996.17,1671.44 1995.4,1661.41 1988.88,1653.92 1979.28,1652.01 1970.53,1656.48 1966.19,1665.49 1968.06,1675.37 1975.35,1682.05 1985.1,1682.82 \n 1993.3,1677.34 1996.58,1667.87 1993.61,1658.3 1985.6,1652.54 1975.83,1652.98 1968.33,1659.42 1966.19,1669.23 1970.27,1678.37 1978.91,1683.06 1988.55,1681.38 \n 1995.21,1674.02 1996.13,1664.01 1990.95,1655.49 1981.8,1651.97 1972.46,1654.91 1966.79,1663.1 1967.14,1673.15 1973.37,1680.9 1982.9,1683.13 1991.77,1678.92 \n 1996.31,1670.02 1994.64,1660.12 1987.43,1653.33 1977.7,1652.47 1969.48,1657.9 1966.19,1667.37 1969.21,1676.93 1977.28,1682.6 1987.03,1682.02 1994.4,1675.43 \n 1996.32,1665.57 1991.98,1656.56 1983.19,1652.17 1973.63,1654.23 1967.29,1661.88 1966.83,1671.92 1972.44,1680.15 1981.78,1683.12 1990.92,1679.58 1996.04,1671.02 \n 1994.99,1661.03 1988.2,1653.8 1978.53,1652.37 1970.04,1657.35 1966.31,1666.64 1968.9,1676.33 1976.73,1682.34 1986.49,1682.14 1994.08,1675.81 1996.29,1666.02 \n 1992.18,1656.9 1983.49,1652.3 1973.89,1654.18 1967.45,1661.72 1966.88,1671.76 1972.43,1680.03 1981.75,1683.03 1990.89,1679.49 1995.98,1670.91 1994.84,1660.93 \n 1987.98,1653.78 1978.29,1652.51 1969.9,1657.65 1966.37,1667.02 1969.22,1676.64 1977.22,1682.4 1986.97,1681.85 1994.32,1675.23 1996.12,1665.35 1991.61,1656.44 \n 1982.71,1652.3 1973.23,1654.69 1967.21,1662.61 1967.24,1672.65 1973.3,1680.53 1982.8,1682.87 1991.66,1678.66 1996.1,1669.71 1994.2,1659.86 1986.77,1653.33 \n 1977.01,1652.93 1969.12,1658.84 1966.48,1668.51 1970.23,1677.79 1978.76,1682.67 1988.4,1681.06 1994.99,1673.65 1995.68,1663.63 1990.19,1655.33 1980.87,1652.33 \n 1971.76,1655.92 1966.78,1664.57 1968.11,1674.52 1975.15,1681.48 1984.87,1682.42 1993.06,1676.94 1996.15,1667.41 1992.82,1657.97 1984.5,1652.71 1974.81,1653.92 \n 1967.96,1661.07 1966.92,1671.06 1972.15,1679.54 1981.38,1682.82 1990.58,1679.47 1995.75,1670.95 1994.63,1660.97 1987.71,1653.89 1978.01,1652.79 1969.76,1658.17 \n 1966.57,1667.66 1969.84,1677.12 1978.13,1682.42 1987.83,1681.22 1994.67,1674.06 1995.67,1664.07 1990.38,1655.63 1981.12,1652.45 1971.96,1655.91 1966.92,1664.51 \n 1968.22,1674.47 1975.28,1681.4 1985,1682.25 1993.11,1676.66 1996.02,1667.08 1992.46,1657.73 1983.99,1652.73 1974.35,1654.3 1967.8,1661.74 1967.22,1671.77 \n 1972.89,1679.95 1982.28,1682.66 1991.25,1678.71 1995.83,1669.84 1993.98,1659.98 1986.53,1653.5 1976.77,1653.28 1969.04,1659.41 1966.77,1669.17 1970.97,1678.24 \n 1979.78,1682.57 1989.27,1680.24 1995.23,1672.29 1994.99,1662.25 1988.68,1654.6 1979.09,1652.75 1970.5,1657.52 1966.75,1666.79 1969.52,1676.42 1977.56,1682.1 \n 1987.28,1681.3 1994.35,1674.37 1995.58,1664.42 1990.43,1655.89 1981.21,1652.61 1972.05,1656.05 1967.05,1664.67 1968.47,1674.61 1975.66,1681.39 1985.4,1681.97 \n 1993.31,1676.1 1995.83,1666.4 1991.82,1657.25 1983.1,1652.76 1973.58,1654.96 1967.56,1662.86 1967.76,1672.89 1974.08,1680.54 1983.68,1682.33 1992.21,1677.47 \n 1995.83,1668.15 1992.88,1658.58 1984.71,1653.1 1975.01,1654.18 1968.18,1661.35 1967.32,1671.34 1972.81,1679.64 1982.17,1682.47 1991.14,1678.54 1995.66,1669.65 \n 1993.66,1659.82 1986.05,1653.54 1976.3,1653.66 1968.84,1660.13 1967.08,1670 1971.81,1678.77 1980.89,1682.44 1990.16,1679.34 1995.4,1670.88 1994.21,1660.92 \n 1987.13,1654.02 1977.4,1653.33 1969.48,1659.17 1966.98,1668.87 1971.06,1677.99 1979.85,1682.33 1989.32,1679.92 1995.11,1671.85 1994.58,1661.84 1987.96,1654.47 \n 1978.3,1653.14 1970.03,1658.46 1966.97,1667.99 1970.52,1677.33 1979.06,1682.19 1988.64,1680.32 1994.84,1672.58 1994.81,1662.55 1988.56,1654.85 1978.97,1653.05 \n 1970.47,1657.97 1967,1667.35 1970.17,1676.84 1978.5,1682.05 1988.14,1680.56 1994.62,1673.07 1994.94,1663.04 1988.94,1655.13 1979.41,1653.02 1970.78,1657.68 \n 1967.05,1666.95 1969.97,1676.52 1978.18,1681.93 1987.85,1680.68 1994.48,1673.33 1994.98,1663.31 1989.12,1655.3 1979.62,1653.03 1970.93,1657.56 1967.1,1666.79 \n 1969.92,1676.39 1978.08,1681.87 1987.76,1680.68 1994.43,1673.36 1994.96,1663.35 1989.1,1655.33 1979.61,1653.07 1970.92,1657.63 1967.13,1666.86 1970.01,1676.44 \n 1978.21,1681.87 1987.88,1680.58 1994.46,1673.18 1994.86,1663.16 1988.89,1655.24 1979.36,1653.14 1970.76,1657.86 1967.15,1667.18 1970.24,1676.69 1978.56,1681.9 \n 1988.19,1680.36 1994.58,1672.78 1994.69,1662.76 1988.48,1655.03 1978.89,1653.25 1970.46,1658.28 1967.19,1667.72 1970.62,1677.11 1979.14,1681.97 1988.7,1680.01 \n 1994.75,1672.16 1994.42,1662.14 1987.87,1654.72 1978.2,1653.43 1970.03,1658.9 1967.25,1668.5 1971.18,1677.68 1979.95,1682.04 1989.38,1679.51 1994.96,1671.3 \n 1994.03,1661.32 1987.03,1654.35 1977.31,1653.71 1969.51,1659.73 1967.38,1669.51 1971.93,1678.37 1980.99,1682.07 1990.2,1678.83 1995.16,1670.2 1993.48,1660.33 \n 1985.96,1653.95 1976.22,1654.14 1968.94,1660.8 1967.63,1670.73 1972.91,1679.15 1982.24,1682.01 1991.14,1677.92 1995.3,1668.86 1992.73,1659.19 1984.65,1653.59 \n 1974.97,1654.75 1968.38,1662.12 1968.04,1672.14 1974.14,1679.96 1983.71,1681.79 1992.13,1676.75 1995.32,1667.28 1991.72,1657.97 1983.09,1653.33 1973.6,1655.62 \n 1967.88,1663.72 1968.69,1673.71 1975.65,1680.72 1985.37,1681.35 1993.12,1675.29 1995.13,1665.48 1990.42,1656.71 1981.28,1653.25 1972.17,1656.79 1967.53,1665.6 \n 1969.63,1675.38 1977.44,1681.36 1987.16,1680.62 1994.02,1673.5 1994.67,1663.51 1988.8,1655.52 1979.27,1653.45 1970.76,1658.31 1967.43,1667.73 1970.93,1677.08 \n 1979.53,1681.77 1989.01,1679.51 1994.72,1671.39 1993.85,1661.41 1986.82,1654.48 1977.09,1654.01 1969.46,1660.23 1967.68,1670.07 1972.63,1678.7 1981.87,1681.85 \n 1990.84,1677.97 1995.12,1668.97 1992.59,1659.3 1984.49,1653.73 1974.84,1655.03 1968.4,1662.54 1968.37,1672.56 1974.76,1680.12 1984.4,1681.47 1992.52,1675.95 \n 1995.09,1666.28 1990.83,1657.28 1981.85,1653.4 1972.62,1656.58 1967.72,1665.23 1969.6,1675.06 1977.31,1681.17 1987.02,1680.53 1993.9,1673.44 1994.51,1663.44 \n 1988.55,1655.52 1978.99,1653.63 1970.6,1658.72 1967.58,1668.23 1971.43,1677.43 1980.23,1681.71 1989.58,1678.93 1994.8,1670.48 1993.27,1660.59 1985.77,1654.21 \n 1976.04,1654.55 1968.97,1661.44 1968.11,1671.41 1973.89,1679.47 1983.4,1681.56 1991.88,1676.63 1995.04,1667.17 1991.31,1657.92 1982.56,1653.55 1973.19,1656.26 \n 1967.93,1664.68 1969.43,1674.57 1976.93,1680.94 1986.66,1680.58 1993.69,1673.66 1994.48,1663.68 1988.61,1655.68 1979.07,1653.72 1970.68,1658.78 1967.68,1668.31 \n 1971.6,1677.47 1980.45,1681.63 1989.75,1678.67 1994.77,1670.09 1992.97,1660.25 1985.26,1654.15 1975.55,1654.88 1968.79,1662.08 1968.41,1672.09 1974.61,1679.8 \n 1984.23,1681.3 1992.37,1675.82 1994.88,1666.15 1990.48,1657.22 1981.41,1653.59 1972.31,1657.12 1967.82,1666 1970.24,1675.7 1978.33,1681.25 1987.96,1679.81 \n 1994.19,1672.13 1993.83,1662.12 1987.07,1654.93 1977.37,1654.24 1969.7,1660.4 1968.01,1670.25 1973.16,1678.74 1982.51,1681.5 1991.25,1677.12 1994.89,1667.83 \n 1991.55,1658.43 1982.95,1653.76 1973.52,1656.21 1968.12,1664.54 1969.54,1674.44 1977.04,1680.81 1986.76,1680.36 1993.67,1673.31 1994.2,1663.32 1988.08,1655.55 \n 1978.47,1654.01 1970.35,1659.5 1967.91,1669.19 1972.43,1678.05 1981.57,1681.49 1990.57,1677.72 1994.79,1668.7 1992.01,1659.11 1983.69,1653.94 1974.15,1655.86 \n 1968.34,1663.89 1969.3,1673.85 1976.52,1680.55 1986.25,1680.5 1993.41,1673.74 1994.27,1663.77 1988.39,1655.81 1978.82,1654 1970.57,1659.3 1967.95,1668.93 \n 1972.32,1677.86 1981.4,1681.43 1990.45,1677.76 1994.72,1668.77 1991.97,1659.17 1983.66,1653.99 1974.12,1655.95 1968.37,1664.02 1969.43,1673.96 1976.74,1680.56 \n 1986.46,1680.35 1993.48,1673.43 1994.11,1663.45 1988.02,1655.66 1978.41,1654.15 1970.34,1659.73 1968.07,1669.46 1972.79,1678.2 1982.02,1681.35 1990.88,1677.24 \n 1994.7,1668.04 1991.45,1658.61 1982.86,1653.94 1973.46,1656.49 1968.23,1664.92 1969.94,1674.77 1977.69,1680.81 1987.36,1679.85 1993.84,1672.39 1993.68,1662.39 \n 1986.96,1655.16 1977.26,1654.54 1969.73,1660.85 1968.37,1670.75 1973.9,1678.97 1983.39,1681.14 1991.78,1676.09 1994.6,1666.52 1990.35,1657.53 1981.29,1653.91 \n 1972.26,1657.6 1968.07,1666.62 1970.97,1676.16 1979.4,1681.13 1988.87,1678.86 1994.3,1670.57 1992.79,1660.69 1985.15,1654.52 1975.46,1655.35 1968.93,1662.75 \n 1969.05,1672.75 1975.76,1679.98 1985.46,1680.56 1992.94,1674.18 1994.18,1664.27 1988.51,1656.15 1978.99,1654.19 1970.72,1659.45 1968.18,1669.1 1972.72,1677.93 \n 1981.9,1681.2 1990.77,1677.13 1994.55,1667.92 1991.19,1658.54 1982.51,1654.06 1973.19,1656.89 1968.28,1665.52 1970.43,1675.27 1978.48,1680.86 1988.07,1679.28 \n 1994.01,1671.36 1993.07,1661.42 1985.77,1654.82 1976.06,1655.17 1969.22,1662.26 1968.95,1672.25 1975.42,1679.71 1985.09,1680.59 1992.72,1674.4 1994.15,1664.51 \n 1988.6,1656.31 1979.09,1654.26 1970.81,1659.48 1968.27,1669.13 1972.85,1677.94 1982.07,1681.1 1990.88,1676.89 1994.47,1667.61 1990.88,1658.32 1982.07,1654.11 \n 1972.86,1657.29 1968.29,1666.11 1970.87,1675.74 1979.18,1680.91 1988.67,1678.78 1994.13,1670.52 1992.57,1660.66 1984.83,1654.61 1975.18,1655.7 1968.91,1663.34 \n 1969.48,1673.31 1976.57,1680.14 1986.28,1680.06 1993.26,1673.12 1993.67,1663.14 1987.28,1655.62 1977.61,1654.71 1970,1660.91 1968.65,1670.81 1974.31,1678.93 \n 1983.85,1680.78 1991.99,1675.35 1994.24,1665.63 1989.34,1657 1980.01,1654.25 1971.41,1658.9 1968.32,1668.36 1972.45,1677.41 1981.5,1681 1990.46,1677.16 \n 1994.35,1668 1991,1658.63 1982.28,1654.24 1973.03,1657.29 1968.41,1666.08 1970.99,1675.71 1979.33,1680.82 1988.78,1678.56 1994.08,1670.19 1992.26,1660.38 \n 1984.35,1654.6 1974.75,1656.09 1968.83,1664.01 1969.91,1673.93 1977.37,1680.31 1987.05,1679.58 1993.53,1672.14 1993.17,1662.16 1986.17,1655.23 1976.47,1655.24 \n 1969.49,1662.19 1969.17,1672.17 1975.67,1679.58 1985.36,1680.26 1992.77,1673.82 1993.76,1663.89 1987.75,1656.04 1978.13,1654.71 1970.32,1660.64 1968.72,1670.49 \n 1974.23,1678.7 1983.75,1680.66 1991.9,1675.24 1994.08,1665.51 1989.06,1656.96 1979.67,1654.43 1971.23,1659.35 1968.5,1668.93 1973.04,1677.76 1982.27,1680.83 \n 1990.97,1676.4 1994.2,1666.99 1990.14,1657.93 1981.07,1654.35 1972.16,1658.3 1968.46,1667.53 1972.08,1676.8 1980.95,1680.83 1990.04,1677.34 1994.17,1668.3 \n 1990.99,1658.87 1982.31,1654.42 1973.07,1657.48 1968.55,1666.31 1971.32,1675.87 1979.81,1680.71 1989.16,1678.06 1994.04,1669.44 1991.65,1659.76 1983.36,1654.57 \n 1973.92,1656.84 1968.72,1665.27 1970.75,1675.03 1978.84,1680.53 1988.36,1678.62 1993.85,1670.39 1992.15,1660.56 1984.24,1654.78 1974.67,1656.36 1968.92,1664.41 \n 1970.32,1674.28 1978.05,1680.31 1987.67,1679.02 1993.63,1671.16 1992.51,1661.24 1984.95,1655 1975.3,1656.02 1969.14,1663.73 1970.01,1673.66 1977.43,1680.1 \n 1987.1,1679.31 1993.43,1671.75 1992.76,1661.8 1985.49,1655.2 1975.8,1655.79 1969.33,1663.22 1969.81,1673.18 1976.97,1679.91 1986.66,1679.49 1993.26,1672.18 \n 1992.92,1662.2 1985.86,1655.37 1976.17,1655.65 1969.48,1662.87 1969.69,1672.85 1976.68,1679.76 1986.37,1679.6 1993.13,1672.44 1993.01,1662.46 1986.08,1655.49 \n 1976.38,1655.58 1969.58,1662.69 1969.64,1672.66 1976.54,1679.68 1986.23,1679.63 1993.06,1672.55 1993.02,1662.57 1986.14,1655.54 1976.45,1655.58 1969.62,1662.66 \n 1969.66,1672.63 1976.55,1679.65 1986.24,1679.6 1993.05,1672.5 1992.97,1662.52 1986.06,1655.54 1976.36,1655.64 1969.6,1662.79 1969.74,1672.76 1976.71,1679.69 \n 1986.4,1679.5 1993.1,1672.29 1992.86,1662.32 1985.82,1655.47 1976.13,1655.78 1969.52,1663.07 1969.89,1673.03 1977.02,1679.78 1986.7,1679.33 1993.2,1671.93 \n 1992.68,1661.98 1985.43,1655.35 1975.76,1655.99 1969.39,1663.51 1970.11,1673.45 1977.48,1679.92 1987.14,1679.07 1993.34,1671.41 1992.41,1661.49 1984.89,1655.2 \n 1975.26,1656.29 1969.24,1664.11 1970.43,1674 1978.11,1680.08 1987.71,1678.72 1993.5,1670.73 1992.04,1660.87 1984.18,1655.04 1974.63,1656.7 1969.08,1664.87 \n 1970.86,1674.68 1978.9,1680.24 1988.39,1678.24 1993.66,1669.88 1991.54,1660.15 1983.31,1654.89 1973.9,1657.25 1968.95,1665.81 1971.43,1675.45 1979.85,1680.37 \n 1989.16,1677.62 1993.77,1668.85 1990.9,1659.33 1982.28,1654.79 1973.09,1657.95 1968.87,1666.92 1972.15,1676.3 1980.97,1680.43 1990,1676.83 1993.8,1667.66 \n 1990.08,1658.46 1981.08,1654.78 1972.23,1658.84 1968.89,1668.19 1973.07,1677.18 1982.24,1680.38 1990.87,1675.84 1993.71,1666.31 1989.06,1657.57 1979.73,1654.91 \n 1971.37,1659.94 \n \"/>\n<circle clip-path=\"url(#clip9305)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1981.33\" cy=\"1667.6\" r=\"18\"/>\n<circle clip-path=\"url(#clip9305)\" style=\"fill:#e26f46; stroke:none; fill-opacity:1\" cx=\"1981.33\" cy=\"1667.6\" r=\"14\"/>\n</svg>\n" | |
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| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### 数式処理系による定積分の計算" | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = symbols(\"x\", positive=true)\nI1 = sympy.Integral(x^(-1)*sin(x), (x,0,oo))\nld(I1, \" = \", I1.doit())", | |
| "execution_count": 15, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{\\sin{\\left (x \\right )}}{x}\\, dx = \\frac{\\pi}{2}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{\\sin{\\left (x \\right )}}{x}\\, dx = \\frac{\\pi}{2}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "a, x = symbols(\"a, x\", positive=true)\nI1 = sympy.Integral(exp(-a*x)*x^(-1)*sin(x), (x,0,oo))\nld(I1, \" = \", I1.doit())", | |
| "execution_count": 16, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{e^{- a x} \\sin{\\left (x \\right )}}{x}\\, dx = - \\operatorname{atan}{\\left (a \\right )} + \\frac{\\pi}{2}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{e^{- a x} \\sin{\\left (x \\right )}}{x}\\, dx = - \\operatorname{atan}{\\left (a \\right )} + \\frac{\\pi}{2}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = symbols(\"x\", positive=true)\n\nI1 = sympy.Integral(cos(x^2), (x,-oo,oo))\nI2 = sympy.Integral(sin(x^2), (x,-oo,oo))\n\nld(I1, \" = \", I1.doit())\nld(I2, \" = \", I2.doit())", | |
| "execution_count": 17, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{-\\infty}^{\\infty} \\cos{\\left (x^{2} \\right )}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$\"", | |
| "text/latex": "$$\\int_{-\\infty}^{\\infty} \\cos{\\left (x^{2} \\right )}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{-\\infty}^{\\infty} \\sin{\\left (x^{2} \\right )}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$\"", | |
| "text/latex": "$$\\int_{-\\infty}^{\\infty} \\sin{\\left (x^{2} \\right )}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = symbols(\"x\", positive=true)\n\nI1 = sympy.Integral(x^(-Sym(1)/2)*cos(x), (x,0,oo))\nI2 = sympy.Integral(x^(-Sym(1)/2)*sin(x), (x,0,oo))\n\nld(I1, \" = \", I1.doit())\nld(I2, \" = \", I2.doit())", | |
| "execution_count": 18, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{\\cos{\\left (x \\right )}}{\\sqrt{x}}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{\\cos{\\left (x \\right )}}{\\sqrt{x}}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{\\sin{\\left (x \\right )}}{\\sqrt{x}}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{\\sin{\\left (x \\right )}}{\\sqrt{x}}\\, dx = \\frac{\\sqrt{2} \\sqrt{\\pi}}{2}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Dirichlet積分の分母を $x^a$ とする一般化\n\n**問題:** $a>0$ のとき, 広義積分\n\n$$\n\\int_1^\\infty \\frac{\\sin x}{x^a}\\,dx = \\lim_{R\\to\\infty}\\int_1^R \\frac{\\sin x}{x^a}\\,dx\n$$\n\nが収束することを示せ. \n\n**解答例:** 部分積分によって\n\n$$\n\\begin{aligned}\n\\int_1^R \\frac{\\sin a}{x^a}\\,dx &= \n\\int_1^R x^{-a}(-\\cos x)'\\,dx \n\\\\ &=\n[-x^{-a}\\cos x]_1^R -\\int_1^R (x^{-a})'(-\\cos x)\\,dx\n\\\\ &=\n-R^{-a}\\cos R+\\cos 1 - a\\int_1^R x^{-a-1}\\cos x\\,dx.\n\\end{aligned}\n$$\n\n第1項の $-R^{-a}\\cos R$ は $R\\to\\infty$ で $0$ に収束し, 第3項の積分は\n\n$$\n\\int_1^R |x^{-a-1}\\cos x|\\,dx \\leqq\n\\int_1^R x^{-a-1}\\,dx =\n\\left[\\frac{x^{-a}}{-a}\\right]_1^R =\n\\frac{1-R^{-a}}{a}\n$$\n\nなので $R\\to\\infty$ で絶対収束する. これで示すべきことがすべて示された. $\\QED$\n\n**解説:** \n$$\n\\int_1^R x^{-a}\\,dx = \\begin{cases}\n\\dfrac{1-R^{1-a}}{1-a} & (a\\ne 1) \\\\\n\\log R & (a=1)\n\\end{cases}\n$$\n\nなので, $a\\leqq 1$ のとき, この積分は $R\\to\\infty$ で無限大に発散する. しかし, $x^{-a}$ を微分すると, $-a x^{-(a+1)}$ になり, $a>0$ ならば $a+1>1$ となり, $x^{-(a+1)}$ の $1$ から $\\infty$ までの積分が収束することを使える. 上の問題の解答例では, 部分積分によってそれを実現している. 絶対収束しない積分を上手に部分積分すると絶対収束する形に直せることがよくある. 部分積分はこの意味での非常に有用である. $\\QED$" | |
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| "cell_type": "markdown", | |
| "source": "#### 部分積分による広義積分の収束の証明法\n\n**問題:** $a(x)$, $b(x)$ は $x\\geqq 1$ の連続函数であるとし, $A(x)=\\int_1^x a(\\xi)\\,d\\xi$ とおく. $A(x)$ は $x\\geqq 1$ で有界であると仮定する: $A(x)\\leqq M$ ($x\\geqq 1$). さらに $b(x)$ は単調減少な $C^1$ 函数でかつ $x\\to\\infty$ で $b(x)\\to 0$ となっていると仮定する(特に $b(x)\\geqq 0$ ($x\\geqq 0$) である). このとき, 広義積分\n\n$$\n\\int_1^\\infty a(x)b(x)\\,dx = \\lim_{R\\to\\infty}\\int_1^R a(x)b(x)\\,dx\n\\tag{1}\n$$\n\nが収束することを示せ.\n\n**解答例:** 部分積分によって,\n\n$$\n\\int_1^R a(x)b(x)\\,dx = A(R)b(R) - A(1)b(1) - \\int_1^R A(x) b'(x)\\,dx.\n\\tag{2}\n$$\n\nそして, $x\\geqq 1$ のとき, $|A(x)|\\leqq M$ かつ $b(x)$ の単調減少性より $b'(x)\\leqq 0$ なので,\n\n$$\n|A(x)b'(x)|\\leqq -Mb'(x) \\quad (x\\geqq 1).\n$$\n\nゆえに, $b(x)\\geqq 0$ であることも使うと, \n\n$$\n\\int_1^R |A(x) b'(x)|\\,dx \\leqq -\\int_1^R M b'(x)\\,dx = M(b(1)-b(R)) \\leqq M b(1) < \\infty.\n$$\n\nこれで, 上の(2)の右辺の積分が絶対収束することがわかった. (2)の右辺の $A(R)b(R)$ は $R\\to\\infty$ のとき $b(R)\\to 0$ より $0$ に収束する. これで, (1)の広義積分が収束することがわかった. $\\QED$\n\n**注意:** 上の解答例と\n\n* 黒木玄, <a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/02+series.ipynb\">02 級数</a> の <a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/02+series.ipynb#%E6%9D%A1%E4%BB%B6%E5%8F%8E%E6%9D%9F%E3%81%99%E3%82%8B%E8%A4%87%E7%B4%A0%E7%B4%9A%E6%95%B0%E3%81%AE%E4%BE%8B\">2.2 条件収束する複素級数の例</a>\n\nにある解説を比較してみよ. 積分と和の違いに過ぎず, 本質的には同じ内容であることがわかる. $\\QED$" | |
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| "source": "## 積分と極限の交換\n\nこのノートを含む一連のノート群では安全な極限の交換しか扱わない.\n\n読者は積分と極限の交換, 積分と微分の交換, 積分順序の交換などを形式的に自由にできることを認めて読んで欲しい.\n\nもしくは以下の結果が証明抜きで自由に使用されると考えてもよい.\n\n**(J)** **積分の順序交換:** $\\ds\\int_a^b\\left(\\int_c^d |f(x,y)| \\,dy\\right)\\,dx < \\infty$ ならば\n\n$$\n\\int_a^b\\left(\\int_c^d f(x,y) \\,dy\\right)\\,dx = \\int_c^d\\left(\\int_a^b f(x,y) \\,dx\\right)\\,dy.\n$$\n\n**(K)** **積分記号化での微分:** 各パラメーター $\\alpha$ について $\\int_a^b|f(\\alpha,x)|\\,dx <\\infty$ であり, 各 $x\\in(a,b)$ について偏導函数(偏微分に値を持つ函数)\n\n$$\n\\frac{\\d f(\\alpha, x)}{\\d\\alpha} = \\lim_{h\\to 0}\\frac{f(\\alpha+h,x)-f(\\alpha,x)}{h}\n$$\n\nが存在し, ある非負値函数 $\\varphi(x)\\geqq 0$ でかつ\n$$\n\\int_a^b \\varphi(x)\\,dx<\\infty, \\qquad\n\\left|\\frac{\\d f(\\alpha,x)}{\\d\\alpha}\\right| \\leqq \\varphi(x)\\quad (a<x<b)\n$$\n\nを満たすものが存在するならば,\n\n$$\n\\frac{d}{d\\alpha}\\int_a^b f(\\alpha,x)\\,dx = \\int_a^b \\frac{\\d f(\\alpha,x)}{\\d\\alpha}\\,dx.\n$$\n\n**(L)** **Lebesgue(ルベーグ)の収束定理:** 函数列 $f_n(x)$ は各点収束している(すなわち各 $x\\in(a,b)$ ごとに数列 $f_n(x)$ が収束している)とし, ある非負値函数 $\\varphi(x)\\geqq 0$ で\n\n$$\n\\int_a^b \\varphi(x)\\,dx <\\infty\n$$\n\nを満たすものが存在すると仮定する. このとき, \n\n$$\n\\lim_{n\\to\\infty}\\int_a^b f_n(x)\\,dx = \\int_a^b \\lim_{n\\to\\infty} f_n(x)\\,dx.\n$$\n\n以上は少々荒い説明の仕方になっている. 正確でかつより一般的な結果およびその証明については\n\n* 猪狩惺, 『<a href=\"https://www.amazon.co.jp/dp/4000054449\">実解析入門</a>』, 岩波書店, 1996\n\nなど多くの教科書に書いてある. この本は品切れで中古本にプレミア価格がついている. 大学の図書館などで借りて読めば安上がりである. その本において, (L)はp.78の定理3.23に, (K)はp.79の例3.3に, (J)のより正確な表現であるFubiniの定理はpp.120-123に書いてある." | |
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| "source": "### 積分と極限が交換できない場合\n\nこのノートを含む一連のノート群では安全な極限の交換しか行わないが, 積分と極限が交換できない例は非常に簡単に作れる. 以下を見よ." | |
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| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Lebesgueの収束定理の結論が成立しない場合1\n\n**例(Lebesgueの収束定理の結論が成立しない場合1):** 函数列 $f_n(x)$ を\n\n$$\nf_n(x) = \\begin{cases}\n1/n & (0< x \\leqq n) \\\\\n0 & (\\text{otherwise}) \\\\\n\\end{cases}\n$$\n\nと定める. さらに函数 $g(x)$ を\n\n$$\ng(x) = \\max\\{f_n(x)\\}_{n=1}^\\infty = \\begin{cases}\n1/n & (n-1<x\\leqq n) \\\\\n0 & (\\text{otherwise}) \\\\ \n\\end{cases}\n$$\n\nと定める. このとき, 任意の実数 $x$ について $\\ds\\lim_{n\\to\\infty}f_n(x)= 0$ だが, $\\ds\\int_0^\\infty f_n(x)\\,dx = 1$ なので, \n\n$$\n\\ds \\lim_{n\\to\\infty}\\int_0^\\infty f_n(x)\\,dx=1\\ne0=\\int_0^\\infty \\lim_{n\\to\\infty}f_n(x)\\,dx\n$$\n\nとなる. そして, \n\n$$\n\\int_0^\\infty g(x)\\,dx = \\sum_{n=1}^\\infty\\frac{1}{n} = \\infty\n$$\n\nであることにも注意せよ. $\\QED$\n\n**問題:** 上の問題の $f_n(x)$, $g(x)$ のグラフを描け.\n\n**解答例:** 次のセルをみよ. $\\QED$" | |
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| "source": "f(n,x) = 0 < x ≤ n ? 1/n : zero(x)\ng(x) = x > 0 ? 1/ceil(x) : zero(x)\nx = 0.0:0.02:10.0\n\nPP = []\nfor n in 1:8\n P = plot(x, f.(n,x), label=\"f_$n(x)\")\n push!(PP, P)\nend\nP = plot(x, g.(x), label=\"g(x)\")\npush!(PP, P)\n#pngplot(P..., size=(750, 600), ylims=(0,1))\nplot(PP..., size=(750, 600), ylims=(0,1))", | |
| "execution_count": 19, | |
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| "source": "### Dirichlet積分とその一般化\n\n条件収束する広義積分に関する有名な公式\n\n$$\n\\int_0^\\infty \\frac{\\sin x}{x}\\,dx = \\frac{\\pi}{2}\n$$\n\nの左辺はよく**Dirichlet積分**と呼ばれる.\n\nDirichlet積分の公式の様々な証明については以下の文献を参照せよ.\n\n* D. H. Hardy, The Integral $\\int_0^\\infty \\frac{\\sin x}{x}\\,dx$, The Mathematical Gazette Vol. 5, No. 80 (Jun. - Jul., 1909), pp. 98-103. ([JSTOR](https://www.jstor.org/stable/3602798))\n\n* A. C. Dixon, Proof That $\\int_0^\\infty \\frac{\\sin x}{x}\\,dx=\\frac{\\pi}{2}$, The Mathematical Gazette Vol. 6, No. 96 (Jan., 1912), pp. 223-224. ([JSTOR](https://www.jstor.org/stable/3604314))\n\nこの一連のノート群ではDirichlet積分の公式の導出を何度も繰り返し扱うが, そこで使った方法の多くはこの2つの文献で紹介されている方法の範囲に含まれている.\n\nDirichlet積分の公式は, ノート「<a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/12+Fourier+analysis.ipynb\">12 Fourier解析</a>」でFourier変換の逆変換の収束性に関するDiniの条件を扱うときに用いられる." | |
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| "source": "#### Dirichlet積分の一般化に関する公式\n\n**問題:** Dirichlet積分の一般化に関する公式\n\n$$\n\\int_0^\\infty \\frac{e^{-ax}\\sin x}{x}\\,dx = \\int_a^\\infty \\frac{dt}{1+t^2} = \\frac{\\pi}{2} - \\arctan a\n\\quad (a\\geqq 0)\n$$\n\nを $a>0$ の場合について示せ.\n\n**注意:** 上の問題の公式の $a\\searrow 0$ の極限としてDirichlet積分の公式が得られる." | |
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| "source": "**解答例1:** $a>0$ と仮定し, $\\ds I(a) = \\int_0^\\infty \\frac{e^{-ax}\\sin x}{x}\\,dx$ とおく.\n\n$a\\to\\infty$ で $I(a)\\to 0$ である($I(\\infty)=0$ とおく). 積分記号化での微分(K)と上の方にある三角函数のLaplace変換の問題の結果より\n\n$$\nI'(a) = \\int_0^\\infty \\frac{\\d}{\\d a}\\frac{e^{-ax}\\sin x}{x}\\,dx = -\\int_0^\\infty e^{-ax}\\sin x\\,dx =\n\\frac{-1}{1+a^2}.\n$$\n\nゆえに\n\n$$\nI(a) = I(\\infty) + \\int_\\infty^a \\frac{-1}{1+t^2}\\,dt = \\int_a^\\infty \\frac{dt}{1+t^2}.\n$$\n\nさらに\n\n$$\n\\int_a^\\infty \\frac{dt}{1+t^2} = \n\\int_0^\\infty \\frac{dt}{1+t^2} - \\int_0^a \\frac{dt}{1+t^2} =\n\\frac{\\pi}{2} - \\arctan a.\n$$\n\nこれで示すべきことがすべて示された. $\\QED$" | |
| }, | |
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| "source": "**解答例2:** $a>0$ であると仮定する. $x>0$ のとき, $\\ds \\frac{e^{-ax}}{x} = \\int_a^\\infty e^{-tx}\\,dt$ なので\n\n$$\n\\int_0^\\infty \\frac{e^{-ax}\\sin x}{x}\\,dx =\n\\int_0^\\infty \\left(\\int_a^\\infty e^{-tx}\\sin x\\,dt\\right)\\,dx.\n$$\n\n積分の順序交換(J)によって, \n\n$$\n\\int_0^\\infty \\left(\\int_a^\\infty e^{-tx}\\sin x\\,dt\\right)\\,dx =\n\\int_a^\\infty \\left(\\int_0^\\infty e^{-tx}\\sin x\\,dx\\right)\\,dt.\n$$\n\n上の方の三角函数のLaplace変換の問題の結果より,\n\n$$\n\\int_a^\\infty \\left(\\int_0^\\infty e^{-tx}\\sin x\\,dx\\right)\\,dt =\n\\int_a^\\infty \\frac{1}{1+t^2}\\,dt.\n$$\n\nこれで $a>0$ の場合の公式(2)が示された. $\\QED$\n\n**注意:** $a>0$ の場合には絶対収束する積分しか出て来ないので積分順序を自由に交換できて簡単である. $a=0$ の場合にも積分順序の交換が可能なことを示せれば $a=0$ の場合の(1)も証明される. $\\QED$" | |
| }, | |
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| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Dirichlet積分の公式の証明のための準備\n\n**問題(次の問題の解答例1のための準備):** 次の公式を示せ: $t>0$ のとき,\n\n$$\n\\int e^{-tx} \\cos x\\,dx = \\frac{-e^{-tx}}{1+t^2}(t\\cos x-\\sin x), \\quad\n\\int e^{-tx} \\sin x\\,dx = \\frac{-e^{-tx}}{1+t^2}(t\\sin x+\\cos x).\n$$\n\n**解答例:** \n$$\n\\begin{aligned}\n\\int e^{-tx}e^{ix}\\,dx &= \n\\int e^{-(t-i)x}\\,dx =\n\\frac{e^{-(t-i)x}}{-(t-i)} =\n-\\frac{t+i}{t^2+1}e^{-tx}(\\cos x+i\\sin x) \n\\\\ &=\n\\frac{-e^{-tx}}{1+t^2}((t\\cos x-\\sin x) + i(t\\sin x+\\cos x)).\n\\end{aligned}\n$$\n\nこの等式の両辺の実部と虚部が示すべき公式にちょうどなっている. $\\QED$\n\nこの問題の結果より, $t>0$ のとき, \n\n$$\n\\begin{aligned}\n&\n\\int_0^R e^{-tx}\\cos x\\,dx = \\frac{-e^{-tR}}{1+t^2}(t\\cos R-\\sin R) + \\frac{t}{1+t^2},\n\\\\ &\n\\int_0^R e^{-tx}\\sin x\\,dx = \\frac{-e^{-tR}}{1+t^2}(t\\sin R+\\cos R) + \\frac{1}{1+t^2}.\n\\end{aligned}\n$$" | |
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| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "t, x = symbols(\"t x\", positive=true)\n\nI1 = sympy.Integral(exp(-t*x)*cos(x), x)\nI2 = sympy.Integral(exp(-t*x)*sin(x), x)\n\nld(I1, \" = \", I1.doit().simplify())\nld(I2, \" = \", I2.doit().simplify())", | |
| "execution_count": 20, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int e^{- t x} \\cos{\\left (x \\right )}\\, dx = \\frac{\\left(- t \\cos{\\left (x \\right )} + \\sin{\\left (x \\right )}\\right) e^{- t x}}{t^{2} + 1}$$\"", | |
| "text/latex": "$$\\int e^{- t x} \\cos{\\left (x \\right )}\\, dx = \\frac{\\left(- t \\cos{\\left (x \\right )} + \\sin{\\left (x \\right )}\\right) e^{- t x}}{t^{2} + 1}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
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| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int e^{- t x} \\sin{\\left (x \\right )}\\, dx = - \\frac{\\left(t \\sin{\\left (x \\right )} + \\cos{\\left (x \\right )}\\right) e^{- t x}}{t^{2} + 1}$$\"", | |
| "text/latex": "$$\\int e^{- t x} \\sin{\\left (x \\right )}\\, dx = - \\frac{\\left(t \\sin{\\left (x \\right )} + \\cos{\\left (x \\right )}\\right) e^{- t x}}{t^{2} + 1}$$" | |
| }, | |
| "metadata": {} | |
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| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Dirichlet積分の公式の証明\n\n**問題:** Dirichlet積分の公式\n\n$$\n\\int_0^\\infty \\frac{\\sin x}{x}\\,dx = \\lim_{R\\to\\infty}\\int_0^R\\frac{\\sin x}{x}\\,dx = \\frac{\\pi}{2}\n$$\n\nを示せ. (1つ目の等号は広義積分の定義であり, 示す必要はない.)\n\n**解答例1:** $R>0$ であるとする. $x>0$ のとき, $\\ds \\frac{1}{x} = \\int_0^\\infty e^{-tx}\\,dt$ なので\n\n$$\n\\int_0^R \\frac{\\sin x}{x}\\,dx =\n\\int_0^R \\left(\\int_0^\\infty e^{-tx}\\sin x\\,dt\\right)\\,dx.\n$$\n\n積分の順序交換(J)によって\n\n$$\n\\int_0^R \\frac{\\sin x}{x}\\,dx =\n\\int_0^\\infty \\left(\\int_0^R e^{-tx}\\sin x\\,dx\\right)\\,dt.\n$$\n\nすぐ上の問題の結果より, \n\n$$\n\\int_0^R \\frac{\\sin x}{x}\\,dx =\n\\int_0^\\infty\\frac{-e^{-tR}}{1+t^2}(t\\sin R+\\cos R)\\,dt + \n\\int_0^\\infty \\frac{dt}{1+t^2}.\n$$\n\n右辺の前者の積分は $R\\to\\infty$ で $0$ に収束し, 後者の積分は $\\pi/2$ に等しいので,\n\n$$\n\\int_0^\\infty \\frac{\\sin x}{x}\\,dx =\n\\lim_{R\\to\\infty}\\int_0^R \\frac{\\sin x}{x}\\,dx =\n\\frac{\\pi}{2}.\n$$\n\nこれで公式(1)が示された. $\\QED$" | |
| }, | |
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| "cell_type": "markdown", | |
| "source": "**解答例2:** $N$ は正の整数であるとする.\n\n$$\n\\sum_{k=-N}^N e^{2\\pi ikx} = e^{-2\\pi iNx}\\frac{e^{2\\pi i(2N+1)x}-1}{e^{2\\pi ix}-1} =\n\\frac{e^{\\pi i(2N+1)x}-e^{-\\pi i(2N+1)x}}{e^{\\pi ix}-e^{-\\pi ix}} =\n\\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}\n$$\n\nなので, $N\\to\\infty$ のとき,\n\n$$\n\\begin{aligned}\n1 &= \\int_{-1/2}^{1/2} \\sum_{k=-N}^N e^{2\\pi ikx}\\,dx =\n\\int_{-1/2}^{1/2} \\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}\\,dx =\n2\\int_0^{1/2} \\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}\\,dx \n\\\\ &=\n\\frac{2}{\\pi} \\int_0^{\\pi(2N+1)/2} \\frac{\\sin t}{(2N+1)\\sin\\left(\\frac{t}{2N+1}\\right)}\\,dt\n\\to\n\\frac{2}{\\pi} \\int_0^\\infty \\frac{\\sin t}{t}\\,dt.\n\\end{aligned}\n$$\n\n4つ目の等号で $\\ds x = \\frac{t}{\\pi(2N+1)}$ とおいた. $N\\to\\infty$ の極限操作では $\\ds(2N+1)\\sin\\left(\\frac{t}{2N+1}\\right)\\to t$ を用いた.\n\nゆえに $\\ds \\int_0^\\infty \\frac{\\sin t}{t}\\,dt = \\frac{\\pi}{2}$. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**解答例2':** 上の解答例2で $N\\to\\infty$ での極限を雑に扱った部分を, Fourier解析におけるRiemann-Lebesgueの定理を認めて使うことによって, より正確に扱ってみよう.\n\n$\\ds\\frac{1}{\\sin(\\pi x)} - \\frac{1}{\\pi x}$ は $x=0$ まで連続函数として延長され, $[-1/2,1/2]$ 上の連続函数になる. ゆえにRiemann-Lebesgueの定理より, $N\\to\\infty$ のとき\n\n$$\n\\eps_N :=\n2\\int_0^{1/2} \\left(\\frac{1}{\\sin(\\pi x)} - \\frac{1}{\\pi x}\\right)\\sin(\\pi(2N+1)x)\\,dx \\to 0.\n$$\n\n一方, $\\ds x = \\frac{t}{\\pi(2N+1)}$ とおくと, $N\\to\\infty$ のとき\n\n$$\n2\\int_0^{1/2} \\frac{\\sin(\\pi(2N+1)x)}{\\pi x}\\,dx =\n\\frac{2}{\\pi} \\int_0^{\\pi(2N+1)/2}\\frac{\\sin t}{t}\\,dt \\to \n\\frac{2}{\\pi} \\int_0^\\infty \\frac{\\sin t}{t}\\,dt.\n$$\n\nゆえに, $N\\to\\infty$ のとき, 途中まで解答例1と同様にして, \n\n$$\n\\begin{aligned}\n1 &= \\int_{-1/2}^{1/2} \\sum_{k=-N}^N e^{2\\pi ikx}\\,dx =\n\\int_{-1/2}^{1/2} \\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}\\,dx =\n2\\int_0^{1/2} \\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}\\,dx \n\\\\ &=\n\\eps_N + 2\\int_0^{1/2} \\frac{\\sin(\\pi(2N+1)x)}{\\pi x}\\,dx\n\\to\n\\frac{2}{\\pi} \\int_0^\\infty \\frac{\\sin t}{t}\\,dt\n\\end{aligned}\n$$\n\nゆえに $\\ds \\int_0^\\infty \\frac{\\sin t}{t}\\,dt = \\frac{\\pi}{2}$. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": " ### Fresnel積分の特殊値の公式の導出\n \n #### Fresnel積分の特殊値の公式の証明のための準備 (1/(1+t⁴)の積分)\n \n **問題:** 次を示せ.\n\n$$\n\\int_0^\\infty \\frac{dt}{1+t^4} = \n\\int_0^\\infty \\frac{t^2\\,dt}{1+t^4} =\n\\frac{\\sqrt{2}\\;\\pi}{4}\n$$" | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "t = symbols(\"t\", real=true)\n\nI1 = sympy.Integral(t^2/(t^4+1), (t,0,oo))\nI2 = sympy.Integral(1/(t^4+1), (t,0,oo))\n\nld(I1, \" = \", I1.doit().simplify())\nld(I2, \" = \", I2.doit().simplify())", | |
| "execution_count": 21, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{t^{2}}{t^{4} + 1}\\, dt = \\frac{\\sqrt{2} \\pi}{4}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{t^{2}}{t^{4} + 1}\\, dt = \\frac{\\sqrt{2} \\pi}{4}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{1}{t^{4} + 1}\\, dt = \\frac{\\sqrt{2} \\pi}{4}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{1}{t^{4} + 1}\\, dt = \\frac{\\sqrt{2} \\pi}{4}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**解答例1:** $t=1/u$ とおくと,\n\n$$\n\\int_0^\\infty \\frac{dt}{1+t^4} = \\int_\\infty^0 \\frac{-u^{-2}\\,du}{1+u^{-4}} =\n\\int_0^\\infty \\frac{u^2\\,du}{u^4+1} = \\int_0^\\infty\\frac{t^2\\,dt}{1+t^4}.\n$$\n\n天下りになってしまうが, $F(t)$ を次のように定める:\n\n$$\nF(t) = \\frac{\\sqrt{2}}{8}\\log\\frac{t^2+\\sqrt{2}\\;t+1}{t^2-\\sqrt{2}\\;t+1}+\n\\frac{\\sqrt{2}}{4}\\left(\n\\arctan\\left(\\sqrt{2}\\;t+1\\right) + \\arctan\\left(\\sqrt{2}\\;t-1\\right)\n\\right).\n$$\n\nこれの導函数を計算すると $F'(t)=1/(1+t^4)$ となることを確認できる(かなり面倒な計算が必要になる). したがって, \n\n$$\n\\int_0^\\infty \\frac{dt}{1+t^4} = [F(t)]_0^\\infty = F(\\infty) = \\frac{\\sqrt{2}}{4}\\pi.\n\\qquad\\QED\n$$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "$1/(1+t^4)$ の原始函数の求め方については\n\n* E. ハイラー, G. ワナー, 『<a href=\"https://www.amazon.co.jp/dp/4621062034\">解析教程 上</a>』\n\nの第2章(5.16)式の周辺(pp.172-174)が詳しい. $1/(1+t^4)$ の部分分数展開を使う. 下の方の解答例2の直後の問題とその解答例も見よ." | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "t = symbols(\"t\")\nr = sqrt(Sym(2))\nH = r/8*log((t^2+r*t+1)/(t^2-r*t+1)) + r/4*(atan(r*t+1)+atan(r*t-1))\nld(\"F(t) = \", H)\nld(raw\"\\frac{d}{dt}F(t) = \", simplify(diff(H, t)))", | |
| "execution_count": 22, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$F(t) = \\frac{\\sqrt{2} \\left(\\operatorname{atan}{\\left (\\sqrt{2} t - 1 \\right )} + \\operatorname{atan}{\\left (\\sqrt{2} t + 1 \\right )}\\right)}{4} + \\frac{\\sqrt{2} \\log{\\left (\\frac{t^{2} + \\sqrt{2} t + 1}{t^{2} - \\sqrt{2} t + 1} \\right )}}{8}$$\"", | |
| "text/latex": "$$F(t) = \\frac{\\sqrt{2} \\left(\\operatorname{atan}{\\left (\\sqrt{2} t - 1 \\right )} + \\operatorname{atan}{\\left (\\sqrt{2} t + 1 \\right )}\\right)}{4} + \\frac{\\sqrt{2} \\log{\\left (\\frac{t^{2} + \\sqrt{2} t + 1}{t^{2} - \\sqrt{2} t + 1} \\right )}}{8}$$" | |
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| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\frac{d}{dt}F(t) = \\frac{1}{t^{4} + 1}$$\"", | |
| "text/latex": "$$\\frac{d}{dt}F(t) = \\frac{1}{t^{4} + 1}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**解答例2:** 複素解析を使う方法. $R>0$ とし, $C_1$ は $0$ から $R$ への線分 $t$, $0\\leqq t\\leqq R$ であるとし, $C_2$ は $R$ から $iR$ への曲線 $Re^{i\\theta}$, $0\\leqq\\theta\\leqq\\pi/2$, $C_3$ は $Ri$ から $0$ への線分であるとする. このとき, \n\n$$\n\\int_{C_3}\\frac{dz}{1+z^4} = -\\int_0^R\\frac{d(it)}{1+(it)^4} =\n-i\\int_0^R \\frac{dt}{1+t^4} = -i\\int_{C_1}\\frac{dt}{1+t^4}.\n$$\n\n$1/(1+z^4)$ の $z=e^{i\\pi/4}$ における留数は\n\n$$\n\\frac{1}{(e^{i\\pi/4}-e^{i3\\pi/4})(e^{i\\pi/4}-e^{i5\\pi/4})(e^{i\\pi/4}-e^{i7\\pi/4})} =\n\\frac{1}{\\sqrt{2}\\cdot\\sqrt{2}(1+i)\\cdot \\sqrt{2}\\;i} =\n\\frac{1}{2\\sqrt{2}\\;(1+i)i}\n$$\n\nなので, 留数定理より,\n\n$$\n\\frac{2\\pi i}{2\\sqrt{2}\\;(1+i)i} = (1-i)\\int_{C_1}\\frac{dz}{1+z^4} + \\int_{C_2}\\frac{dz}{1+z^4}.\n$$\n\n$R\\to\\infty$ で右辺の $C_1$ 上での積分は $\\int_0^\\infty dt/(1+t^4)$ に収束し, $C_2$ 上での積分は $0$ に収束するので,\n\n$$\n\\frac{2\\pi i}{2\\sqrt{2}\\;(1+i)i} = (1-i)\\int_0^\\infty\\frac{dt}{1+t^4}.\n$$\n\nゆえに,\n\n$$\n\\int_0^\\infty\\frac{dt}{1+t^4} =\n\\frac{2\\pi}{2\\sqrt{2}\\cdot 2} =\n\\frac{\\sqrt{2}\\;\\pi}{4}.\n\\qquad\\QED\n$$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**問題(上の解答例1で使った結果):** $1/(t^4+1)$ の原始函数を求めよ.\n\n**解答例:** 以下, 記号の簡単のため $r=\\sqrt{2}$ とおく($r^2=2$). 因数分解\n\n$$\nt^4+1 = (t^2+rt + 1)(t^2-rt+1)\n$$\n\nより, \n\n$$\n\\frac{2r}{t^4+1} = \\frac{t+r}{t^2+rt+1}-\\frac{t-r}{t^2-rt+1}.\n$$\n\n一般に $a>0$ のとき\n\n$$\n\\int\\frac{s\\;ds}{s^2+a^2} = \\frac{1}{2}\\log(s^2+a^2), \\quad\n\\int\\frac{ds}{s^2+a^2} = \\frac{1}{a}\\arctan\\frac{s}{a}.\n$$\n\n$t=s-r/2$ とおくと\n\n$$\n\\begin{aligned}\n\\int\\frac{t+r}{t^2+rt+1}\\,dt &= \\int\\frac{s+r/2}{s^2+1/2}\\,ds =\n\\frac{1}{2}\\log(2s^2+1) + \\arctan(rs)\n\\\\ &=\n\\frac{1}{2}\\log 2 + \\frac{1}{2}\\log(t^2+rt+1) + \\arctan(rt+1).\n\\end{aligned}\n$$\n\n$t=s+r/2$ とおくと\n\n$$\n\\begin{aligned}\n-\\int \\frac{t-r}{t^2-rt+1}\\,dt &= -\\int \\frac{s-r/2}{s^2+1/2}\\,ds =\n-\\frac{1}{2}\\log(2s^2+1) + \\arctan(rs)\n\\\\ &=\n-\\frac{1}{2}\\log 2 - \\frac{1}{2}\\log(t^2-rt+1) + \\arctan(rt-1).\n\\end{aligned}\n$$\n\nゆえに\n\n$$\n2r\\int\\frac{dt}{t^4+1} =\n\\frac{1}{2}\\log\\frac{t^2+rt+1}{t^2-rt+1} + \\arctan(rt+1) + \\arctan(rt-1).\n$$\n\nしたがって,\n\n$$\n\\int\\frac{dt}{t^4+1} = \n\\frac{1}{4r}\\log\\frac{t^2+rt+1}{t^2-rt+1} + \n\\frac{1}{2r}(\\arctan(rt+1) + \\arctan(rt-1)).\n\\qquad \\QED\n$$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### 1/(1+t⁴)の不定積分の公式の数式処理を用いた確認" | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "s = symbols(\"s\", real=true)\na = symbols(\"a\", positive=true)\n\nI1 = sympy.Integral(s/(s^2+a^2), s)\nI2 = sympy.Integral(1/(s^2+a^2), s)\n\nld(I1, \" = \", I1.doit())\nld(I2, \" = \", I2.doit())", | |
| "execution_count": 23, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int \\frac{s}{a^{2} + s^{2}}\\, ds = \\frac{\\log{\\left (a^{2} + s^{2} \\right )}}{2}$$\"", | |
| "text/latex": "$$\\int \\frac{s}{a^{2} + s^{2}}\\, ds = \\frac{\\log{\\left (a^{2} + s^{2} \\right )}}{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int \\frac{1}{a^{2} + s^{2}}\\, ds = \\frac{\\operatorname{atan}{\\left (\\frac{s}{a} \\right )}}{a}$$\"", | |
| "text/latex": "$$\\int \\frac{1}{a^{2} + s^{2}}\\, ds = \\frac{\\operatorname{atan}{\\left (\\frac{s}{a} \\right )}}{a}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "r, s, t = symbols(\"r s t\", real=true)\nv = sqrt(Sym(2))\n\nld(\"r = \", v)\n\nA = (t + r)/(t^2 + r*t + 1)\nB = -(t - r)/(t^2 - r*t + 1)\nH = expand(1/((t^2 + r*t + 1)*(t^2 - r*t + 1)))\n\nld(\"A = \", A)\nld(\"B = \", B)\nld(\"H = \", H(r=>v))\nld(\"A + B - 2rH = \", simplify(A+B-2r*H)) # Verify 2r*H == A+B\n\n# integrate A\nAs = simplify(simplify(A(t=>s-r/2))(r^2=>2))\nintAs = integrate(s/(s^2+Sym(1)/2), s) + integrate(r/2/(s^2+Sym(1)/2), s)\nintA = simplify(expand(intAs(s=>t+r/2)))\n\nld(raw\"A|_{t=s-r/2} = \", As)\nld(raw\"\\int A|_{t=s-r/2}\\,ds = \", simplify(intAs(r=>v)))\nld(raw\"\\int A\\,dt = \", simplify(intA(r=>v)))\n\n# integrate B\nBs = simplify(simplify(B(t=>s+r/2))(r^2=>2))\nintBs = -integrate(s/(s^2+Sym(1)/2), s) + integrate(r/2/(s^2+Sym(1)/2), s)\nintB = simplify(expand(intBs(s=>t-r/2)))\n\nld(raw\"B|_{t=s+r/2} = \", Bs)\nld(raw\"\\int B|_{t=s+r/2}\\,ds = \", simplify(intBs(r=>v)))\nld(raw\"\\int B\\,dt = \", simplify(intB(r=>v)))\n\nK = simplify((1/(2r)*(intA + intB))(r=>v))\n\nld(raw\"K(t) = \\frac{1}{2r}\\left(\\int A\\,dt + \\int B\\,dt\\right)\")\nld(raw\" = \", K)\nld(raw\"\\frac{d}{dt}K(t) = \", simplify(diff(K, t))) # Verification of dF/dt = 1/(t^4+1)", | |
| "execution_count": 24, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$r = \\sqrt{2}$$\"", | |
| "text/latex": "$$r = \\sqrt{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$A = \\frac{r + t}{r t + t^{2} + 1}$$\"", | |
| "text/latex": "$$A = \\frac{r + t}{r t + t^{2} + 1}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$B = \\frac{r - t}{- r t + t^{2} + 1}$$\"", | |
| "text/latex": "$$B = \\frac{r - t}{- r t + t^{2} + 1}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$H = \\frac{1}{t^{4} + 1}$$\"", | |
| "text/latex": "$$H = \\frac{1}{t^{4} + 1}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$A + B - 2rH = 0$$\"", | |
| "text/latex": "$$A + B - 2rH = 0$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$A|_{t=s-r/2} = \\frac{r + 2 s}{2 s^{2} + 1}$$\"", | |
| "text/latex": "$$A|_{t=s-r/2} = \\frac{r + 2 s}{2 s^{2} + 1}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int A|_{t=s-r/2}\\,ds = \\frac{\\log{\\left (2 s^{2} + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} s \\right )}$$\"", | |
| "text/latex": "$$\\int A|_{t=s-r/2}\\,ds = \\frac{\\log{\\left (2 s^{2} + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} s \\right )}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int A\\,dt = \\frac{\\log{\\left (t^{2} + \\sqrt{2} t + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} t + 1 \\right )} + \\frac{\\log{\\left (2 \\right )}}{2}$$\"", | |
| "text/latex": "$$\\int A\\,dt = \\frac{\\log{\\left (t^{2} + \\sqrt{2} t + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} t + 1 \\right )} + \\frac{\\log{\\left (2 \\right )}}{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$B|_{t=s+r/2} = \\frac{r - 2 s}{2 s^{2} + 1}$$\"", | |
| "text/latex": "$$B|_{t=s+r/2} = \\frac{r - 2 s}{2 s^{2} + 1}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int B|_{t=s+r/2}\\,ds = - \\frac{\\log{\\left (2 s^{2} + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} s \\right )}$$\"", | |
| "text/latex": "$$\\int B|_{t=s+r/2}\\,ds = - \\frac{\\log{\\left (2 s^{2} + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} s \\right )}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\int B\\,dt = - \\frac{\\log{\\left (t^{2} - \\sqrt{2} t + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} t - 1 \\right )} - \\frac{\\log{\\left (2 \\right )}}{2}$$\"", | |
| "text/latex": "$$\\int B\\,dt = - \\frac{\\log{\\left (t^{2} - \\sqrt{2} t + 1 \\right )}}{2} + \\operatorname{atan}{\\left (\\sqrt{2} t - 1 \\right )} - \\frac{\\log{\\left (2 \\right )}}{2}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$K(t) = \\frac{1}{2r}\\left(\\int A\\,dt + \\int B\\,dt\\right)$$\"", | |
| "text/latex": "$$K(t) = \\frac{1}{2r}\\left(\\int A\\,dt + \\int B\\,dt\\right)$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$ = \\frac{\\sqrt{2} \\left(- \\log{\\left (t^{2} - \\sqrt{2} t + 1 \\right )} + \\log{\\left (t^{2} + \\sqrt{2} t + 1 \\right )} + 2 \\operatorname{atan}{\\left (\\sqrt{2} t - 1 \\right )} + 2 \\operatorname{atan}{\\left (\\sqrt{2} t + 1 \\right )}\\right)}{8}$$\"", | |
| "text/latex": "$$ = \\frac{\\sqrt{2} \\left(- \\log{\\left (t^{2} - \\sqrt{2} t + 1 \\right )} + \\log{\\left (t^{2} + \\sqrt{2} t + 1 \\right )} + 2 \\operatorname{atan}{\\left (\\sqrt{2} t - 1 \\right )} + 2 \\operatorname{atan}{\\left (\\sqrt{2} t + 1 \\right )}\\right)}{8}$$" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "L\"$$\\frac{d}{dt}K(t) = \\frac{1}{t^{4} + 1}$$\"", | |
| "text/latex": "$$\\frac{d}{dt}K(t) = \\frac{1}{t^{4} + 1}$$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
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| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Fresnel積分の特殊値の公式の証明\n\n**問題:** Fresnel積分の特殊値の公式\n\n$$\n\\int_0^\\infty \\frac{\\cos x}{\\sqrt{x}}\\,dx = \n\\int_0^\\infty \\frac{\\sin x}{\\sqrt{x}}\\,dx = \n\\sqrt{\\frac{\\pi}{2}}\n$$\n\nを示せ." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "**解答例:** Gauss積分の公式より, $x>0$ のとき\n\n$$\n2\\int_0^\\infty e^{-x t^2}\\,dt = \\int_{-\\infty}^\\infty e^{-x t^2}\\,dt = \\sqrt{\\frac{\\pi}{x}}\n$$\n\nゆえに\n\n$$\n\\int_0^\\infty \\frac{e^{ix}}{\\sqrt{x}}\\,dx =\n\\frac{2}{\\sqrt{\\pi}}\\int_0^\\infty\\left(\\int_0^\\infty e^{-x t^2}e^{ix}\\,dt\\right)\\,dx\n$$\n\n積分順序を交換すると(本当はこの部分を上の問題と同様の方法で正当化する必要がある),\n\n$$\n\\frac{2}{\\sqrt{\\pi}}\\int_0^\\infty\\left(\\int_0^\\infty e^{-x t^2}e^{ix}\\,dt\\right)\\,dx =\n\\frac{2}{\\sqrt{\\pi}}\\int_0^\\infty\\left(\\int_0^\\infty e^{-x t^2}e^{ix}\\,dx\\right)\\,dt.\n$$\n\n\n三角函数のLaplace変換の問題の解答例2より, \n\n$$\n\\frac{2}{\\sqrt{\\pi}}\\int_0^\\infty\\left(\\int_0^\\infty e^{-x t^2}e^{ix}\\,dx\\right)\\,dt =\n\\frac{2}{\\sqrt{\\pi}}\\int_0^\\infty\\frac{t^2+i}{t^4+1}\\,dt\n$$\n\n上の問題の結果より,\n\n$$\n\\frac{2}{\\sqrt{\\pi}}\\int_0^\\infty\\frac{t^2+i}{t^4+1}\\,dt =\n\\frac{2}{\\sqrt{\\pi}}(1+i)\\frac{\\sqrt{2}\\;\\pi}{4} =\n(1+i)\\sqrt{\\frac{\\pi}{2}}.\n$$\n\nしたがって,\n\n$$\n\\int_0^\\infty \\frac{e^{ix}}{\\sqrt{x}}\\,dx = (1+i)\\sqrt{\\frac{\\pi}{2}}.\n$$\n\n両辺の実部と虚部を見れば示したい結果が得られる. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "上の解答例と本質的に同じ解答の詳しい解説が\n\n* E. ハイラー, G. ワナー, 『<a href=\"https://www.amazon.co.jp/dp/4621061909\">解析教程 下</a>』\n\nの第4章の演習問題5.14(p.254, この問題はこの本の最後に載っている問題)の解答解説(pp.313-314)にある." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "#### Fresnel積分の特殊値の公式の複素解析を用いた導出\n\n**注意:** 複素解析を使えばFresnel積分の特殊値の公式\n\n$$\n\\int_0^\\infty \\cos t^2\\,dt = \n\\int_0^\\infty \\sin t^2\\,dt = \n\\frac{1}{2}\\sqrt{\\frac{\\pi}{2}}\n$$\n\nを以下のように比較的容易に示すことができる. \n\n\n$R>0$ であるとし, $C_1$ は $0$ から $R$ への線分, $C_2$ は $R$ から $R e^{\\pi i/4}$ への円弧, $C_3$ は $0$ から $R e^{\\pi i/4}$ への線分とすると, Cauchyの積分定理より,\n\n$$\n\\int_{C_3} e^{-z^2}\\,dz = \\int_{C_1} e^{-z^2}\\,dz + \\int_{C_2} e^{-z^2}\\,dz.\n$$\n\nそして,\n\n$$\n\\begin{aligned}\n&\n\\int_{C_3} e^{-z^2}\\,dz = \\int_0^R \\exp(-(e^{\\pi i/4}t)^2)d(e^{\\pi i/4}t)\\,dt = \ne^{\\pi i/4}\\int_0^R e^{-it^2}\\,dt,\n\\\\ &\n\\int_{C_1} e^{-z^2}\\,dz = \\int_0^R e^{-t^2}\\,dt \\to \\frac{\\sqrt{\\pi}}{2}\\quad (R\\to\\infty),\n\\\\ &\n\\int_{C_2} e^{-z^2}\\,dz \\to 0 \\quad (R\\to\\infty).\n\\end{aligned}\n$$\n\nしたがって,\n\n$$\ne^{\\pi i/4} \\int_0^\\infty e^{-it^2}\\,dt = \\frac{\\sqrt{\\pi}}{2}.\n$$\n\n両辺に $\\ds e^{-\\pi i/4} = \\frac{1-i}{\\sqrt{2}}$ をかけると,\n\n$$\n\\int_0^\\infty e^{-it^2}\\,dt = (1-i)\\frac{1}{2}\\sqrt{\\frac{\\pi}{2}}.\n$$\n\n$e^{-it^2}=\\cos t^2 - i\\sin t^2$ に注意して, 両辺の実部と虚部を比較すれば欲しい結果が得られる. $\\QED$" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "上の複素解析を用いたFresnel積分の特殊値の導出のより詳しい解説が\n\n* 高木貞治, 『<a href=\"https://www.amazon.co.jp/dp/4000051717\">解析概論</a>』\n\nの第62節の例2(pp.224-225)にある." | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "ガンマ函数との関連については\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160501StirlingFormula.pdf\">ガンマ分布の中心極限定理とStirlingの公式</a>\n\nの第8.6節を参照せよ. 一般に, ちょっとした複素解析の利用によって, $-1<s<1$ のとき\n\n$$\n\\int_0^\\infty x^{s-1}\\sin x\\,dx = \\Gamma(s)\\sin\\frac{\\pi s}{2} = \\Gamma(s+1)\\frac{\\sin(\\pi s/2)}{s}\n$$\n\nとなることを示せる. ($\\sin(\\pi s/2)/s$ の $s=0$ での値は極限を取ることによって自然に $\\pi/2$ だと定義される.) これの $s=0$ の場合が $\\ds\\int_0^\\infty\\frac{\\sin x}{x}\\,dx=\\frac{\\pi}{2}$ であり, $s=1/2$ の場合が $\\ds\\int_0^\\infty\\frac{\\sin x}{\\sqrt{x}}\\,dx = \\sqrt{\\frac{\\pi}{2}}$ である. このように, Dirichlet積分とFresnel積分(の特殊値)はガンマ函数によって統一的に取り扱うことができる. 別々に公式を証明する必要はない. $\\QED$" | |
| }, | |
| { | |
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| "source": "#### sine integral and cosine integral functions\n\n函数 $\\Si x$, $\\si x$, $\\Ci x$, $\\Cin x$ が以下のように定義される:\n\n$$\n\\begin{alignedat}{2}\n&\n\\Si x = \\int_0^x \\frac{\\sin t}{t}\\,dt,\n& \\quad &\n\\si x = -\\int_x^\\infty \\frac{\\sin t}{t}\\, dt,\n\\\\ &\n\\Ci x = -\\int_x^\\infty \\frac{\\cos t}{t}\\,dt,\n& \\quad &\n\\Cin x = -\\int_0^x \\frac{\\cos t - 1}{t}\\,dt.\n\\end{alignedat}\n$$\n\nこれらのあいだには以下のような関係がある:\n\n$$\n\\begin{aligned}\n&\n\\Si x - \\si x = \\int_0^\\infty\\frac{\\sin t}{t}\\,dt = \\frac{\\pi}{2},\n\\\\ &\n\\Ci x = \\gamma + \\log x - \\Cin x.\n\\end{aligned}\n$$\n\n前者はDirichlet積分の公式そのものである. 後者の公式中の $\\gamma$ はEuler定数 $\\gamma=0.5772\\cdots$ である. 以下では後者を示そう.\n\n(1) $0<s<1$ のとき, \n\n$$\n\\int_0^\\infty t^{s-1} e^{it}\\,dt = e^{\\pi i s/2}\\Gamma(s) =\n\\frac{e^{\\pi i s/2}}{s}\\Gamma(s+1).\n$$\n\nゆえに, 両辺の実部と虚部を取ることによって, \n\n$$\n\\begin{aligned}\n&\n\\int_0^\\infty t^{s-1}\\cos t \\,dt = \\frac{\\cos(\\pi s/2)}{s}\\Gamma(s+1), \n\\\\ &\n\\int_0^\\infty t^{s-1}\\sin t \\,dt = \\frac{\\sin(\\pi s/2)}{s}\\Gamma(s+1).\n\\end{aligned}\n$$\n\n**証明:** ガンマ函数の定義式\n\n$$\n\\Gamma(s) = \\int_0^\\infty u^{s-1} e^{-u}\\,du\n$$\n\nにおいて, $u = -it = e^{-\\pi i/2}t$ とおくと, \n\n$$\n\\Gamma(s) = e^{-\\pi i s/2}\\int_0^{i\\infty} t^{s-1}e^{it}\\,dt.\n$$\n\nCauchyの積分定理によって, 積分経路を正の虚軸から正の実軸に回転できることを示せる:\n\n$$\n\\Gamma(s) = e^{-\\pi i s/2}\\int_0^\\infty t^{s-1}e^{it}\\,dt.\n$$\n\nゆえに上の(1)の公式が成立する. $\\QED$\n\n(2) 上の(1)における $\\sin t$ を含む公式で, $s\\searrow 0$ とするとDirichlet積分の公式が得られる:\n\n$$\n\\int_0^\\infty \\frac{\\sin t}{t} \\,dt = \\frac{\\pi}{2}.\n$$\n\n(3) ガンマ函数 $\\Gamma(s)$ の $s=1$ での微係数はEuler定数の $-1$ 倍に等しい:\n\n$$\n\\Gamma'(1) = -\\gamma.\n$$\n\n**証明:** ガンマ函数に関するGaussの公式\n\n$$\n\\Gamma(s) = \\lim_{n\\to\\infty}\\frac{n^s n!}{s(s+1)\\cdots(s+n)}.\n$$\n\nの両辺の対数を $s$ について微分すると,\n\n$$\n\\frac{\\Gamma'(s)}{\\Gamma(s)} = \n\\lim_{n\\to\\infty}\\left(\n\\log n -\\left(\\frac{1}{s}+\\frac{1}{s+1}+\\cdots+\\frac{1}{s+n}\\right)\n\\right)\n$$\n\nなので, 特に $s=1$ とおくと, Euler定数の定義より, \n\n$$\n\\Gamma'(1) = \\lim_{n\\to\\infty}\\left(\n\\log n -\\left(\\frac{1}{1}+\\frac{1}{2}+\\cdots+\\frac{1}{n+1}\\right)\n\\right)\n= -\\gamma.\n\\qquad \\QED\n$$\n\n(4) $s\\searrow 0$ において, \n\n$$\n\\int_0^\\infty t^{s-1}\\cos t \\,dt = \\frac{1}{s} - \\gamma + O(s).\n$$\n\n**証明:** 上の(1)の $\\cos t$ が出て来る公式および $\\cos x = 1 + O(x^2)$ と $\\Gamma(1)=1$ と(3) $\\Gamma'(1)=-\\gamma$ より, \n\n$$\n\\begin{aligned}\n\\int_0^\\infty t^{s-1}\\cos t \\,dt &= \\frac{\\cos(\\pi s/2)}{s}\\Gamma(s+1) \n\\\\ & =\n\\frac{1+O(s^2)}{s}(1-\\gamma s + O(s^2)) =\n\\frac{1}{s} - \\gamma + O(s).\n\\qquad \\QED\n\\end{aligned}\n$$\n\n(5) $x>0$ のとき, $s\\searrow 0$ において, \n\n$$\n\\int_0^x t^{s-1}\\cos t\\,dt =\n\\frac{1}{s} + \\log x + \\int_0^x \\frac{\\cos t - 1}{t}\\,dt + O(s).\n$$\n\n**証明:** $t^{s-1}(\\cos t - 1) = O(t^{s+1})$ に注意すれば,\n\n$$\n\\begin{aligned}\n\\int_0^x t^{s-1}\\cos t\\,dt &=\n\\int_0^x t^{s-1}\\,dt + \\int_0^x t^{s-1}(\\cos t - 1)\\,dt\n\\\\ &=\n\\frac{x^s}{s} + \\int_0^x t^s\\frac{\\cos t - 1}{t}\\,dt\n\\\\ &=\n\\frac{1}{s}(1+s\\log x + O(s^2)) + \\int_0^x (1+s\\log t + O(t^2))\\frac{\\cos t - 1}{t}\\,dt\n\\\\ &=\n\\frac{1}{s} + \\log x + \\int_0^x \\frac{\\cos t - 1}{t}\\,dt + O(s).\n\\qquad \\QED\n\\end{aligned}\n$$\n\n(6) $x>0$ のとき,\n\n$$\n\\int_x^\\infty \\frac{\\cos t}{t}\\,dt = -\\gamma -\\log x - \\int_0^x \\frac{\\cos t - 1}{t}\\,dt.\n$$\n\nこれは $\\Ci x = \\gamma + \\log x - \\Cin x$ が成立することを意味している.\n\n**証明:** (4)と(5)より,\n\n$$\n\\begin{aligned}\n\\int_x^\\infty \\frac{\\cos t}{t}\\,dt &=\n\\lim_{s\\searrow 0}\\int_x^\\infty t^{s-1}\\cos t\\,dt\n\\\\ &=\n\\lim_{s\\searrow 0}\\left(\n\\int_0^\\infty t^{s-1}\\cos t\\,dt - \\int_0^x t^{s-1}\\cos t\\,dt\n\\right)\n\\\\ &=\n\\lim_{s\\searrow 0}\\left(\n\\left(\\frac{1}{s}-\\gamma + O(s)\\right) - \n\\left(\\frac{1}{s}+\\log x +\\int_0^x\\frac{\\cos t -1}{t}\\,dt + O(s)\\right)\n\\right)\n\\\\ &=\n-\\gamma -\\log x -\\int_0^x\\frac{\\cos t -1}{t}\\,dt.\n\\qquad \\QED\n\\end{aligned}\n$$" | |
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| "source": "**例:** $\\eps,a>0$ のとき, $t=u/a$ とおくと, (6)より, \n\n$$\n\\int_\\eps^\\infty\\frac{\\cos(at)}{t}\\,dt = \n\\int_{a\\eps}^\\infty\\frac{\\cos u}{u}\\,du =\n-\\gamma - \\log a + \\log\\eps - \\int_0^{a\\eps}\\frac{\\cos u - 1}{u}\\,du\n$$\n\nなので, さらに $b>0$ とすると,\n\n$$\n\\int_\\eps^\\infty\\frac{\\cos(at)-\\cos(bt)}{t}\\,dt = -\n\\log a + \\log b - \\int_0^{b\\eps}\\frac{\\cos u - 1}{u}\\,du + \\int_0^{b\\eps}\\frac{\\cos u - 1}{u}\\,du\n$$\n\nなので $\\eps\\searrow 0$ の極限を取ると,\n\n$$\n\\int_0^\\infty\\frac{\\cos(at)-\\cos(bt)}{t}\\,dt = -\\log a + \\log b = - \\log\\frac{a}{b}.\n\\qquad \\QED\n$$" | |
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| "source": "@vars a b t positive=true\nI1 = sympy.Integral((cos(a*t)-cos(b*t))/t, (t, 0, oo))\nld(I1, \"=\", I1.doit())", | |
| "execution_count": 25, | |
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| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{\\cos{\\left (a t \\right )} - \\cos{\\left (b t \\right )}}{t}\\, dt=- \\log{\\left (a \\right )} + \\log{\\left (b \\right )}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{\\cos{\\left (a t \\right )} - \\cos{\\left (b t \\right )}}{t}\\, dt=- \\log{\\left (a \\right )} + \\log{\\left (b \\right )}$$" | |
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| "source": "**例:** $a_1,\\ldots,a_r>0$ かつ $\\alpha_1+\\cdots+\\alpha_r = 0$ のとき, 上の例と同様にして,\n\n$$\n\\int_0^\\infty\\frac{\\alpha_1\\cos(a_1 t)+\\cdots+\\alpha_r\\cos(a_r t)}{t}\\,dt =\n-(\\alpha_1\\log a_1 + \\cdots + \\alpha_r\\log a_r).\n\\qquad \\QED\n$$" | |
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| "source": "@vars a b c d t positive=true\n@vars α β γ\nI1 = sympy.Integral((α*cos(a*t)+β*cos(b*t)+γ*cos(c*t)-(α+β+γ)*cos(d*t))/t, (t, 0, oo))\nld(I1, \"\\\\\\\\ =\", I1.doit())", | |
| "execution_count": 26, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
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| "text/plain": "L\"$$\\int_{0}^{\\infty} \\frac{α \\cos{\\left (a t \\right )} + β \\cos{\\left (b t \\right )} + γ \\cos{\\left (c t \\right )} - \\left(α + β + γ\\right) \\cos{\\left (d t \\right )}}{t}\\, dt\\\\ =- α \\log{\\left (a \\right )} + α \\log{\\left (d \\right )} - β \\log{\\left (b \\right )} + β \\log{\\left (d \\right )} - γ \\log{\\left (c \\right )} + γ \\log{\\left (d \\right )}$$\"", | |
| "text/latex": "$$\\int_{0}^{\\infty} \\frac{α \\cos{\\left (a t \\right )} + β \\cos{\\left (b t \\right )} + γ \\cos{\\left (c t \\right )} - \\left(α + β + γ\\right) \\cos{\\left (d t \\right )}}{t}\\, dt\\\\ =- α \\log{\\left (a \\right )} + α \\log{\\left (d \\right )} - β \\log{\\left (b \\right )} + β \\log{\\left (d \\right )} - γ \\log{\\left (c \\right )} + γ \\log{\\left (d \\right )}$$" | |
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