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Taylor.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Taylor展開\n",
"\n",
"微分方程式の数値解法への応用を目的とし, Taylor展開に慣れ親しむためのノートを作成した.\n",
"\n",
"© 2021 Shuhei Ohno\n",
"<br>Source: https://gist.github.com/ohno\n",
"<br>License: https://opensource.org/licenses/MIT"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Julia Version 1.6.2\n",
"Commit 1b93d53fc4 (2021-07-14 15:36 UTC)\n",
"Platform Info:\n",
" OS: Windows (x86_64-w64-mingw32)\n",
" CPU: Intel(R) Core(TM) i7-4650U CPU @ 1.70GHz\n",
" WORD_SIZE: 64\n",
" LIBM: libopenlibm\n",
" LLVM: libLLVM-11.0.1 (ORCJIT, haswell)\n"
]
}
],
"source": [
"versioninfo()\n",
"# using Pkg\n",
"# Pkg.add(\"Plots\")\n",
"# Pkg.add(\"LaTeXStrings\")\n",
"using Plots\n",
"using LaTeXStrings"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 表記ゆれ\n",
"\n",
"この節は読み飛ばしてよい. Taylorの定理・公式, Taylor展開・級数の呼称には表記ゆれがある. このノートではそれぞれ Taylorの定理, Taylor展開と呼ぶ.\n",
"\n",
"|文献|Taylorの定理|Taylor展開|\n",
"|----|----|----|\n",
"|[志賀浩二『解析入門30講』(朝倉書店, 1988)](http://www.asakura.co.jp/books/isbn/978-4-254-11480-5/) p.72,81|テイラーの定理|テイラー展開|\n",
"|[亀谷俊司『解析学入門』(朝倉書店, 1974)](http://www.asakura.co.jp/books/isbn/978-4-254-11709-7/) pp.156,296|テイラーの定理|テイラー展開|\n",
"|[寺田文行『サイエンスライブラリ理工系の数学 17 新微分積分』(サイエンス社, 1979)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0139-8&y=1979) pp.39,44|テーラーの定理|テーラー級数展開|\n",
"|[笠原晧司『サイエンスライブラリ数学 12 微分積分学』(サイエンス社, 1974)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0108-4&y=1974) p.90|テイラー公式|テイラー公式|\n",
"|[中西襄『サイエンス・パレット 032 微分方程式 物理的発想の解析学』(丸善出版, 2016)](https://www.maruzen-publishing.co.jp/item/?book_no=295095) p.24|なし|テイラー展開|\n",
"|[杉浦光夫『基礎数学2 解析入門Ⅰ』(東京大学出版会, 1980)](http://www.utp.or.jp/book/b302042.html) pp.99,101|テイラーの定理|テイラー展開|\n",
"|[杉浦光夫, 清水英男, 金子晃, 岡本和夫『解析演習』(東京大学出版会, 1989)](http://www.utp.or.jp/book/b302122.html) p.92,|テイラー展開|テイラー級数|\n",
"|[高木貞治『定本 解析概論』(岩波書店, 2010)](https://www.iwanami.co.jp/book/b265489.html) pp.66,70|Taylorの公式|Taylor級数|\n",
"|[寺沢寛一『自然科学者のための 数学概論 (増訂版)』(岩波書店, 1983)](https://www.iwanami.co.jp/book/b265411.html) pp.8,9|Taylorの定理|Taylor級数|\n",
"|このノート|Taylorの定理|Taylor展開|"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 記号\n",
"\n",
"この節も読み飛ばしてよい. Taylor展開の中心には$a$または$x_0$が使われることが多い. なお, このノートではEular法や有限差分法(FDM), 特に陽解法で使いやすい定義を採用する.\n",
"\n",
"|文献|変数|中心|変位|\n",
"|----|----|----|----|\n",
"|[志賀浩二『解析入門30講』(朝倉書店, 1988)](http://www.asakura.co.jp/books/isbn/978-4-254-11480-5/)|$$b$$|$$a$$|$$b-a$$|\n",
"|[亀谷俊司『解析学入門』(朝倉書店, 1974)](http://www.asakura.co.jp/books/isbn/978-4-254-11709-7/)|$$x$$|$$x_0$$|$$x-x_0$$|\n",
"|[寺田文行『サイエンスライブラリ理工系の数学 17 新微分積分』(サイエンス社, 1979)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0139-8&y=1979)|$$b$$|$$a$$|$$b-a$$|\n",
"|[笠原晧司『サイエンスライブラリ数学 12 微分積分学』(サイエンス社, 1974)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0108-4&y=1974)|$$x$$|$$x_0$$|$$x-x_0$$|\n",
"|[中西襄『サイエンス・パレット 032 微分方程式 物理的発想の解析学』(丸善出版, 2016)](https://www.maruzen-publishing.co.jp/item/?book_no=295095)|$$x$$|$$a$$|$$x-a$$|\n",
"|[杉浦光夫『基礎数学2 解析入門Ⅰ』(東京大学出版会, 1980)](http://www.utp.or.jp/book/b302042.html)|$$x$$|$$a$$|$$x-a$$|\n",
"|[杉浦光夫, 清水英男, 金子晃, 岡本和夫『解析演習』(東京大学出版会, 1989)](http://www.utp.or.jp/book/b302122.html)|$$x$$|$$a$$|$$x-a$$|\n",
"|[高木貞治『定本 解析概論』(岩波書店, 2010)](https://www.iwanami.co.jp/book/b265489.html)|$$x$$|$$a$$|$$x-a$$|\n",
"|[寺沢寛一『自然科学者のための 数学概論 (増訂版)』(岩波書店, 1983)](https://www.iwanami.co.jp/book/b265411.html)|$$x+a$$|$$a$$|$$x$$|\n",
"|このノート|$$x+\\Delta x$$|$$x$$|$$\\Delta x$$|"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Taylorの定理\n",
"\n",
"[杉浦光夫『基礎数学2 解析入門Ⅰ』(東京大学出版会, 1980)](http://www.utp.or.jp/book/b302042.html) pp.99,100 によると, Taylorの定理とは以下である. 証明はp.100に譲るが, Cauchyの平均値の定理を用いて証明される.\n",
"\n",
"\n",
"> 区間$[a,x]=I$（または$[x,a]$）で$n$回微分可能な実数値函数$f$に対して, \n",
"\\begin{aligned}\n",
" f(x) = f(a) + \\frac{f'(a)}{1!} (x-a) + \\frac{f''(a)}{2!} (x-a)^2 + \\cdots + \\frac{f^{(n-1)}(a)}{(n-1)!} (x-a)^{n-1} + R_n(x)\n",
"\\end{aligned}\n",
"によって$R_n(x)$を定義するとき\n",
"\\begin{aligned}\n",
" R_n(x) = \\frac{f^{(n)}(c)}{n!} (x-a)^n\n",
"\\end{aligned}\n",
"となる$c\\in I$が存在する.\n",
"\n",
"Taylorの定理も, Cauchyの平均値の定理も, 高校数学でよく見かけた[Lagrangeの平均値の定理](https://ja.wikipedia.org/wiki/%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86)の一般化である. 実際に$x=b$とおいて$n=1$を考えてみると以下のように変形でき, たしかに見覚えがある. 記号の選び方にも納得できるだろう.\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
" f(b) &= f(a) + f'(c) (b-a) \\\\\n",
" \\frac{f(b)-f(a)}{b-a} &= f'(c)\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"<!-- 剰余項$R_n(x)$の表し方は一意ではなく, [笠原晧司『サイエンスライブラリ数学 12 微分積分学』(サイエンス社, 1974)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0108-4&y=1974)では4つの表記(Lagrange, Roche, Cauchy, 積分型)が挙げられている. -->"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Taylor展開\n",
"\n",
"<!--\n",
"高木貞治曰く\n",
"> Taylor級数は解析学において最も重要である.　―　[高木貞治『定本 解析概論』(岩波書店, 2010)](https://www.iwanami.co.jp/book/b265489.html) p.70 より\n",
"-->\n",
"\n",
"以降は独自の記号を採用する. 関数$f$の点$x$周りのTaylor展開は以下である.\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"f(x+\\Delta x)\n",
"&= \\sum_{n=0}^\\infty \\frac{1}{n!} \\frac{{\\rm d}^n f(x)}{{\\rm d}x^n}\\Delta x^n \\\\\n",
"&= f(x)\n",
"+ \\frac{{\\rm d}f(x)}{{\\rm d}x}\\Delta x\n",
"+ \\frac{1}{2} \\frac{{\\rm d}^2f(x)}{{\\rm d}x^2}\\Delta x^2\n",
"+ \\frac{1}{6} \\frac{{\\rm d}^3f(x)}{{\\rm d}x^3}\\Delta x^3\n",
"+ \\frac{1}{24} \\frac{{\\rm d}^4f(x)}{{\\rm d}x^3}\\Delta x^4\n",
"+ \\cdots\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"ただし$0!=1, \\frac{{\\rm d}^0 f(x)}{{\\rm d}x^0}=f(x)$である. $\\Delta x^n$ではなく$(\\Delta x)^n$と書いてもよいが, このノートで$x^n$の増分を考えることはない. Taylor展開が成り立つためには, 区間$[x,x+\\Delta x]\\in I$で$C^\\infty$級で, $I$の各点で$\\lim_{n\\rightarrow\\infty}R_n=0$とならなければならないことは明らかであろう. そのための条件は[杉浦光夫『基礎数学2 解析入門Ⅰ』(東京大学出版会, 1980)](http://www.utp.or.jp/book/b302042.html) p.101を参照されたい. 具体例として$x=0$周りのTaylor展開(Maclaurin展開)の公式を挙げる.\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\sin(\\Delta x) &= \\Delta x - \\frac{1}{3!} \\Delta x^3 + \\frac{1}{5!} \\Delta x^5 - \\frac{1}{7!} \\Delta x^7 + \\cdots \\\\\n",
"\\cos(\\Delta x) &= 1 - \\frac{1}{2!} \\Delta x^2 + \\frac{1}{4!} \\Delta x^4 - \\frac{1}{6!} \\Delta x^6 + \\cdots \\\\\n",
"\\exp(\\Delta x) &= 1 + \\Delta x + \\frac{1}{2!} \\Delta x^2 + \\frac{1}{3!} \\Delta x^3 + \\frac{1}{4!} \\Delta x^4 + \\cdots \\\\\n",
"\\frac{1}{1-\\Delta x} &= 1 + \\Delta x + \\Delta x^2 + \\Delta x^3 + \\Delta x^4 + \\cdots\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"上の3式は$|\\Delta x|<\\infty$で成り立つ(無限大の収束半径をもつ)が, 最後の式は$|\\Delta x|<1$でなければ収束しない. 実際, $\\Delta x = 0.9$のとき, 右辺は$1+0.9+0.81+0.729+\\cdots$と少しずつ小さい値になっていくから, $\\Delta x <1$の条件では収束しそうである. 次に$\\Delta x = 1$のとき, 右辺は$1+1+1+1+\\cdots$なので明らかに発散する. 同様に$\\Delta x = 2$のときも, 右辺は$1+2+4+8+\\cdots$なので明らかに発散する. <b>Taylor展開できたとしても, $\\Delta x$を大きくすると成り立たなくなることがある.</b> 詳しくは[Cauchy–Hadamardの定理](https://ja.wikipedia.org/wiki/%E3%82%B3%E3%83%BC%E3%82%B7%E3%83%BC%E2%80%93%E3%82%A2%E3%83%80%E3%83%9E%E3%83%BC%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86)を参照されたい."
]
},
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"source": [
"## 多変数関数のTaylor展開\n",
"\n",
"偏微分方程式(PDE)の数値解法である有限差分法(FDM)は多変数のTaylor展開を用いて導出される. 例えば, 関数$u(x, t)$の5点$u(x, t+\\Delta t), (x+\\Delta x, t), (x-\\Delta x, t), (x+2\\Delta x, t), (x-2\\Delta x, t)$周りでのTaylor展開は以下である.\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
" u(x, t+\\Delta t)\n",
" &= u (x, t)\n",
" + \\frac{\\partial u(x, t)}{\\partial t} \\Delta t\n",
" + \\frac{1}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial t^{2}} \\Delta t^{2}\n",
" + \\frac{1}{3!} \\frac{\\partial^{3} u(x, t)}{\\partial t^{3}} \\Delta t^{3}\n",
" + \\frac{1}{4!} \\frac{\\partial^{4} u(x, t)}{\\partial t^{4}} \\Delta t^{4}\n",
" + \\cdots\n",
"\\\\\n",
" u(x+\\Delta x, t)\n",
" &= u(x, t)\n",
" + \\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + \\frac{1}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial x^{2}} \\Delta x^{2}\n",
" + \\frac{1}{3!} \\frac{\\partial^{3} u(x, t)}{\\partial x^{3}} \\Delta x^{3}\n",
" + \\frac{1}{4!} \\frac{\\partial^{4} u(x, t)}{\\partial x^{4}} \\Delta x^{4}\n",
" + \\cdots\n",
"\\\\\n",
" u(x-\\Delta x, t)\n",
" &= u(x, t)\n",
" - \\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + \\frac{1}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial x^{2}} \\Delta x^{2}\n",
" - \\frac{1}{3!} \\frac{\\partial^{3} u(x, t)}{\\partial x^{3}} \\Delta x^{3}\n",
" + \\frac{1}{4!} \\frac{\\partial^{4} u(x, t)}{\\partial x^{4}} \\Delta x^{4}\n",
" + \\cdots\n",
"\\\\\n",
" u(x+2\\Delta x, t)\n",
" &= u(x, t)\n",
" + 2\\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + \\frac{2^2}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial x^{2}} \\Delta x^{2}\n",
" + \\frac{2^3}{3!} \\frac{\\partial^{3} u(x, t)}{\\partial x^{3}} \\Delta x^{3}\n",
" + \\frac{2^4}{4!} \\frac{\\partial^{4} u(x, t)}{\\partial x^{4}} \\Delta x^{4}\n",
" + \\cdots\n",
"\\\\\n",
" u(x-2\\Delta x, t)\n",
" &= u(x, t)\n",
" - 2\\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + \\frac{2^2}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial x^{2}} \\Delta x^{2}\n",
" - \\frac{2^3}{3!} \\frac{\\partial^{3} u(x, t)}{\\partial x^{3}} \\Delta x^{3}\n",
" + \\frac{2^4}{4!} \\frac{\\partial^{4} u(x, t)}{\\partial x^{4}} \\Delta x^{4}\n",
" + \\cdots\n",
"\\end{aligned}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 演習 1\n",
"\n",
"$f(x)=\\sin(x)$のTaylor展開を考える. $x=0$周りのTaylor展開(Maclaurin展開)の次数を上げていくごとに, $\\Delta x$が大きくても良い近似を与えるようになることを視覚的に確かめなさい.\n",
"\n",
"## 解答 1"
]
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},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f0(x) = 0\n",
"f1(x) = f0(x) + x\n",
"f2(x) = f1(x) + 0\n",
"f3(x) = f2(x) - x^3/factorial(3)\n",
"f4(x) = f3(x) + 0\n",
"f5(x) = f4(x) + x^5/factorial(5)\n",
"f6(x) = f5(x) + 0\n",
"f7(x) = f6(x) - x^7/factorial(7)\n",
"\n",
"plt = plot(sin, label=\"exact\", xlims=(-3*pi,3*pi), ylims=(-3,3), lw=2, lc=\"#000000\")\n",
"plot!(plt, -2*pi:0.1:2*pi, f1, label=\"n=1\")\n",
"plot!(plt, -2*pi:0.1:2*pi, f3, label=\"n=3\")\n",
"plot!(plt, -2*pi:0.1:2*pi, f5, label=\"n=5\")\n",
"plot!(plt, -2*pi:0.1:2*pi, f7, label=\"n=7\")\n",
"scatter!(plt, [0], [sin(0)], label=\"\")\n",
"xlabel!(plt, \"\\$\\\\Delta x\\$\")\n",
"ylabel!(plt, \"\\$\\\\sin(\\\\Delta x)\\$\")\n",
"plot(plt)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 演習 2\n",
"\n",
"$f(x)=\\sin(x)$を$x=\\frac{\\pi}{2}$周りで4次の項までTaylor展開しなさい.\n",
"\n",
"## 解答 2\n",
"\n",
"4階までの導関数および$x=\\frac{\\pi}{2}$における微分係数はそれぞれ\n",
"$$\n",
"\\begin{aligned}\n",
"f(x) &= \\sin(x) &\n",
"f(\\pi/2) &= 1\\\\\n",
"\\frac{{\\rm d}f(x)}{{\\rm d}x} &= \\cos(x) &\n",
"\\frac{{\\rm d}f(\\pi/2)}{{\\rm d}x} &= 0\\\\\n",
"\\frac{{\\rm d}^2f(x)}{{\\rm d}x^2} &= -\\sin(x) &\n",
"\\frac{{\\rm d}^2f(\\pi/2)}{{\\rm d}x^2} &= -1\\\\\n",
"\\frac{{\\rm d}^3f(x)}{{\\rm d}x^3} &= -\\cos(x) &\n",
"\\frac{{\\rm d}^3f(\\pi/2)}{{\\rm d}x^3} &= 0\\\\\n",
"\\frac{{\\rm d}^4f(x)}{{\\rm d}x^4} &= \\sin(x) &\n",
"\\frac{{\\rm d}^4f(\\pi/2)}{{\\rm d}x^4} &= 1\\\\\n",
"\\end{aligned}\n",
"$$\n",
"である. よって\n",
"$$\n",
"\\sin(\\pi/2 + \\Delta x) \\simeq 1 - \\frac{1}{2!} \\Delta x^2 + \\frac{1}{4!} \\Delta x^4 .\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 演習 3\n",
"\n",
"$f(x)=\\cos(x)$を$x=0$周りで4次の項までTaylor展開しなさい.\n",
"\n",
"## 解答 3\n",
"\n",
"4階までの導関数および$x=\\frac{\\pi}{2}$における微分係数はそれぞれ\n",
"$$\n",
"\\begin{aligned}\n",
"f(x) &= \\cos(x) &\n",
"f(0) &= 1\\\\\n",
"\\frac{{\\rm d}f(x)}{{\\rm d}x} &= -\\sin(x) &\n",
"\\frac{{\\rm d}f(0)}{{\\rm d}x} &= 0\\\\\n",
"\\frac{{\\rm d}^2f(x)}{{\\rm d}x^2} &= -\\cos(x) &\n",
"\\frac{{\\rm d}^2f(0)}{{\\rm d}x^2} &= -1\\\\\n",
"\\frac{{\\rm d}^3f(x)}{{\\rm d}x^3} &= \\sin(x) &\n",
"\\frac{{\\rm d}^3f(0)}{{\\rm d}x^3} &= 0\\\\\n",
"\\frac{{\\rm d}^4f(x)}{{\\rm d}x^4} &= \\cos(x) &\n",
"\\frac{{\\rm d}^4f(0)}{{\\rm d}x^4} &= 1\\\\\n",
"\\end{aligned}\n",
"$$\n",
"である. よって\n",
"$$\n",
"\\cos(\\Delta x) \\simeq 1 - \\frac{1}{2!} \\Delta x^2 + \\frac{1}{4!} \\Delta x^4 .\n",
"$$\n",
"$\\sin(\\pi/2 + \\Delta x)=\\cos(\\Delta x)$が成り立つことが確かめられた."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 演習 4\n",
"\n",
"Taylor展開\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
" u(x+\\Delta x, t)\n",
" &=\n",
" u(x, t)\n",
" + \\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + \\frac{1}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial x^{2}} \\Delta x^{2}\n",
" + O\\left(\\Delta x^{3}\\right)\n",
"\\\\\n",
" u(x-\\Delta x, t)\n",
" &=\n",
" u(x, t)\n",
" - \\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + \\frac{1}{2!} \\frac{\\partial^{2} u(x, t)}{\\partial x^{2}} \\Delta x^{2}\n",
" + O\\left(\\Delta x^{3}\\right)\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"を用いて\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
" \\frac{\\partial u(x, t)}{\\partial x}\n",
" = \n",
" \\frac{u(x+\\Delta x, t)- u(x-\\Delta x, t)}{2\\Delta x}\n",
" + O\\left(\\Delta x^{2}\\right)\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"を示しなさい.\n",
"\n",
"## 解答４\n",
"\n",
"Taylor展開の2式の差を取って変形すればよい.\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
" 2\\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" + O\\left(\\Delta x^{3}\\right)\n",
" &= \n",
" u(x+\\Delta x, t)\n",
" - u(x-\\Delta x, t)\n",
"\\\\\n",
" 2\\frac{\\partial u(x, t)}{\\partial x} \\Delta x\n",
" &= \n",
" u(x+\\Delta x, t)\n",
" - u(x-\\Delta x, t)\n",
" - O\\left(\\Delta x^{3}\\right)\n",
"\\\\\n",
" \\frac{\\partial u(x, t)}{\\partial x}\n",
" &= \n",
" \\frac{u(x+\\Delta x, t)- u(x-\\Delta x, t)}{2\\Delta x}\n",
" - \\frac{O\\left(\\Delta x^{3}\\right)}{2\\Delta x}\n",
"\\\\\n",
" \\frac{\\partial u(x, t)}{\\partial x}\n",
" &= \n",
" \\frac{u(x+\\Delta x, t)- u(x-\\Delta x, t)}{2\\Delta x}\n",
" + O\\left(\\Delta x^{2}\\right)\n",
"\\end{aligned}\n",
"$$"
]
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"## 参考文献\n",
"\n",
"- [寺田文行『サイエンスライブラリ理工系の数学 17 新微分積分』(サイエンス社, 1979)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0139-8&y=1979) pp.39,44\n",
"- [笠原晧司『サイエンスライブラリ数学 12 微分積分学』(サイエンス社, 1974)](https://www.saiensu.co.jp/search/?isbn=978-4-7819-0108-4&y=1974) p.90\n",
"- [中西襄『サイエンス・パレット 032 微分方程式 物理的発想の解析学』(丸善出版, 2016)](https://www.maruzen-publishing.co.jp/item/?book_no=295095) p.24\n",
"- [杉浦光夫『基礎数学2 解析入門Ⅰ』(東京大学出版会, 1980)](http://www.utp.or.jp/book/b302042.html) pp.99,101\n",
"- [杉浦光夫, 清水英男, 金子晃, 岡本和夫『解析演習』(東京大学出版会, 1989)](http://www.utp.or.jp/book/b302122.html) p.92\n",
"- [高木貞治『定本 解析概論』(岩波書店, 2010)](https://www.iwanami.co.jp/book/b265489.html) pp.66,70\n",
"- [寺沢寛一『自然科学者のための 数学概論 (増訂版)』(岩波書店, 1983)](https://www.iwanami.co.jp/book/b265411.html) pp.8,9"
]
}
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