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SpecialFunctions.jl and SpecialPolynomials.jl
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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# 特殊関数\n", | |
| "\n", | |
| "量子力学にはやたらと特殊関数が登場する. 調和振動子や水素原子などの初歩的な系でさえ可視化に苦労したので, Juliaでの特殊関数の利用方法をまとめておくことにした. コメントに要望などを書き込んで頂ければ追記していく." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## パッケージ\n", | |
| "\n", | |
| "Juliaの特殊関数のパッケージは[SpecialFunctions.jl](https://specialfunctions.juliamath.org/stable/functions_overview/)である. ガンマ関数, ベッセル関数, 誤差関数, 楕円積分をはじめとし, 多くの特殊関数はこちらのパッケージで利用可能である. しかし, 量子力学に登場するエルミート多項式, ラゲールの陪多項式, ルジャンドルの陪多項式などはサポートされていないので, [SpecialPolynomials.jl](https://docs.juliahub.com/SpecialPolynomials/)を利用する. これらの多項式は[Polynomials.jl](https://juliamath.github.io/Polynomials.jl/stable/)と連携することで, [微分演算](https://juliamath.github.io/Polynomials.jl/stable/reference/#Polynomials.derivative)や[積分演算](https://juliamath.github.io/Polynomials.jl/stable/reference/#Polynomials.integrate)が可能であり, 文字列として出力することも可能であるから, さらに[Latexify.jl](https://korsbo.github.io/Latexify.jl/stable/)と連携することでLaTeX表示も可能である. 今回はLaTeX表示に少し拘り, [LaTeXStrings](https://github.com/stevengj/LaTeXStrings.jl)も利用した. 他にもPrintf.jl, Plots.jlを使用した." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "# using Pkg\n", | |
| "# Pkg.add(\"Plots\")\n", | |
| "# Pkg.add(\"Printf\")\n", | |
| "# Pkg.add(\"Latexify\")\n", | |
| "# Pkg.add(\"Polynomials\")\n", | |
| "# Pkg.add(\"SpecialPolynomials\")\n", | |
| "# Pkg.add(\"SpecialFunctions\")\n", | |
| "using Plots\n", | |
| "using Printf\n", | |
| "using Latexify\n", | |
| "using LaTeXStrings\n", | |
| "using Polynomials\n", | |
| "using SpecialPolynomials\n", | |
| "using SpecialFunctions" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## LaTeX表示\n", | |
| "\n", | |
| "Latexify.jlに文字列として数式を渡すことで, LaTeX出力してくれる. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 108, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$ax^{2} + bx + c$" | |
| ], | |
| "text/plain": [ | |
| "L\"$ax^{2} + bx + c$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$f\\left( x \\right) = \\frac{1}{2} \\cdot x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$f\\left( x \\right) = \\frac{1}{2} \\cdot x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\mathrm{f\\_1}\\left( x \\right) = \\frac{1}{2} \\cdot x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\mathrm{f\\_1}\\left( x \\right) = \\frac{1}{2} \\cdot x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "latexify(\"ax^2+bx+c\") |> display\n", | |
| "latexify(\"f(x) = 1/2*x\") |> display\n", | |
| "latexify(\"f_1(x) = 1/2*x\") |> display" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "残念なことに$f_1(x)$となってほしいのに$\\mathrm{f\\_1}(x)$となってしまう. LaTeXStrings.jlを利用すると, そのままLaTeXとして解釈されるので今度は式の方がおかしくなってしまう." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 109, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$f_1(x) = 1/2*x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$f_1(x) = 1/2*x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "L\"f_1(x) = 1/2*x\" |> display" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "この問題はLaTeXStrings.jlとLatexify.jlをうまく組み合わせると解決できる. `env=:raw`で`1/2*x`を`\\frac{1}{2} x`に変換し, `latexstring()`に渡すことで結合できる. ついでに`cdot=false`で掛け算の$\\cdot$を非表示にできる." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 110, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "\\frac{1}{2} \\cdot x" | |
| ], | |
| "text/plain": [ | |
| "L\"\\frac{1}{2} \\cdot x\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$f_1(x) = \\frac{1}{2} \\cdot x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$f_1(x) = \\frac{1}{2} \\cdot x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$f_1(x) = \\frac{1}{2} x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$f_1(x) = \\frac{1}{2} x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "latexify(\"1/2*x\", env=:raw) |> display\n", | |
| "latexstring(\"f_1(x) = \", latexify(\"1/2*x\", env=:raw)) |> display\n", | |
| "latexstring(\"f_1(x) = \", latexify(\"1/2*x\", env=:raw, cdot=false)) |> display" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## エルミート多項式\n", | |
| "\n", | |
| "[エルミート多項式](https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%AB%E3%83%9F%E3%83%BC%E3%83%88%E5%A4%9A%E9%A0%85%E5%BC%8F)$H_n(x)$は[調和振動子のSchrödinger方程式の固有関数](https://zenn.dev/ohno/articles/870b0c2a0af590)に登場する. ロドリゲスの公式では\n", | |
| "\n", | |
| "$$\n", | |
| " H_n(x) = (-1)^n \\mathrm{e}^{x^2} \\frac{\\mathrm{d}^n}{\\mathrm{d}x^n} \\mathrm{e}^{-x^2}\n", | |
| "$$\n", | |
| "\n", | |
| "と表される. SpecialPolynomials.jlでサポートされており, $H_n(x)$は`basis(Hermite, n)(x)`のように書けばよい. $n=0\\sim5$の実装が正しいか, LaTeXStrings.jlとLatexify.jlを用いて数式を表示して確認した. [Wikipedia](https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%AB%E3%83%9F%E3%83%BC%E3%83%88%E5%A4%9A%E9%A0%85%E5%BC%8F)と見比べて頂きたい." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 71, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{0}(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{0}(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{1}(x) = 2.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{1}(x) = 2.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{2}(x) = -2.0 + 4.0 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{2}(x) = -2.0 + 4.0 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{3}(x) = -12.0 x + 8.0 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{3}(x) = -12.0 x + 8.0 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{4}(x) = 12.0 - 48.0 x^{2} + 16.0 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{4}(x) = 12.0 - 48.0 x^{2} + 16.0 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{5}(x) = 120.0 x - 160.0 x^{3} + 32.0 x^{5}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{5}(x) = 120.0 x - 160.0 x^{3} + 32.0 x^{5}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{6}(x) = -120.0 + 720.0 x^{2} - 480.0 x^{4} + 64.0 x^{6}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{6}(x) = -120.0 + 720.0 x^{2} - 480.0 x^{4} + 64.0 x^{6}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{7}(x) = -1680.0 x + 3360.0 x^{3} - 1344.0 x^{5} + 128.0 x^{7}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{7}(x) = -1680.0 x + 3360.0 x^{3} - 1344.0 x^{5} + 128.0 x^{7}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{8}(x) = 1680.0 - 13440.0 x^{2} + 13440.0 x^{4} - 3584.0 x^{6} + 256.0 x^{8}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{8}(x) = 1680.0 - 13440.0 x^{2} + 13440.0 x^{4} - 3584.0 x^{6} + 256.0 x^{8}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{9}(x) = 30240.0 x - 80640.0 x^{3} + 48384.0 x^{5} - 9216.0 x^{7} + 512.0 x^{9}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{9}(x) = 30240.0 x - 80640.0 x^{3} + 48384.0 x^{5} - 9216.0 x^{7} + 512.0 x^{9}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{10}(x) = -30240.0 + 302400.0 x^{2} - 403200.0 x^{4} + 161280.0 x^{6} - 23040.0 x^{8} + 1024.0 x^{10}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{10}(x) = -30240.0 + 302400.0 x^{2} - 403200.0 x^{4} + 161280.0 x^{6} - 23040.0 x^{8} + 1024.0 x^{10}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$H_{11}(x) = -665280.0 x + 2.2176e6 x^{3} - 1.77408e6 x^{5} + 506880.0 x^{7} - 56320.0 x^{9} + 2048.0 x^{11}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$H_{11}(x) = -665280.0 x + 2.2176e6 x^{3} - 1.77408e6 x^{5} + 506880.0 x^{7} - 56320.0 x^{9} + 2048.0 x^{11}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:11\n", | |
| " p = basis(Hermite, n)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " latexstring(\"H_{$n}(x) = \", latexify(\"$q\", env=:raw, cdot=false)) |> display\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "例えば$H_2(x) = 4x^2 - 2$なので, $H_2(5) = 98$となるはずである." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 134, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "98.0" | |
| ] | |
| }, | |
| "execution_count": 134, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "p = basis(Hermite, 2)\n", | |
| "p(5)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## ラゲール多項式\n", | |
| "\n", | |
| "注意せよ. 文献によって$1/n!$が有ったり無かったりする. SpecialPolynomials.jlでサポートされているのはロドリゲスの公式\n", | |
| "\n", | |
| "$$\n", | |
| " L_n(x) = \\frac{1}{n!} \\mathrm{e}^x \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left( \\mathrm{e}^{-x} x^n \\right)\n", | |
| "$$\n", | |
| "\n", | |
| "に対応する式であり, $L_n(x)$は`basis(Laguerre{0}, n)(x)`のように書けばよい. [Wikipedia(英語版)](https://en.wikipedia.org/wiki/Laguerre_polynomials#The_first_few_polynomials)と見比べて頂きたい." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 69, | |
| "metadata": { | |
| "scrolled": true | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{0}(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{0}(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{1}(x) = 1.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{1}(x) = 1.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{2}(x) = 1.0 - 2.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{2}(x) = 1.0 - 2.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{3}(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{3}(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{4}(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{4}(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{5}(x) = 1.0 - 5.0 x + 5.0 x^{2} - 1.66667 x^{3} + 0.208333 x^{4} - 0.00833333 x^{5}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{5}(x) = 1.0 - 5.0 x + 5.0 x^{2} - 1.66667 x^{3} + 0.208333 x^{4} - 0.00833333 x^{5}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{6}(x) = 1.0 - 6.0 x + 7.5 x^{2} - 3.33333 x^{3} + 0.625 x^{4} - 0.05 x^{5} + 0.00138889 x^{6}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{6}(x) = 1.0 - 6.0 x + 7.5 x^{2} - 3.33333 x^{3} + 0.625 x^{4} - 0.05 x^{5} + 0.00138889 x^{6}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_{7}(x) = 1.0 - 7.0 x + 10.5 x^{2} - 5.83333 x^{3} + 1.45833 x^{4} - 0.175 x^{5} + 0.00972222 x^{6} - 0.000198413 x^{7}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_{7}(x) = 1.0 - 7.0 x + 10.5 x^{2} - 5.83333 x^{3} + 1.45833 x^{4} - 0.175 x^{5} + 0.00972222 x^{6} - 0.000198413 x^{7}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:7\n", | |
| " p = basis(Laguerre{0}, n)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " latexstring(\"L_{$n}(x) = \", latexify(\"$q\", env=:raw, cdot=false)) |> display\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "$1/n!$について補足する. ラゲール多項式$L_n(x)$はそもそも微分方程式\n", | |
| "\n", | |
| "$$\n", | |
| " xy'' + (1-x)y' + ny = 0\n", | |
| "$$\n", | |
| "\n", | |
| "を満たす解$y$であるから, 定数$1/n!$倍だけ異なっても解であることには違いない. もし\n", | |
| "\n", | |
| "$$\n", | |
| " \\tilde{L}_n(x) = n!\\cdot L_n(x) = \\mathrm{e}^x \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left( \\mathrm{e}^{-x} x^n \\right)\n", | |
| "$$\n", | |
| "\n", | |
| "を利用したいのであれば$L_n(x)$に$n!$を掛ければよいので, `factorial(n)*basis(Laguerre{0}, n)(x)`とすればよい. [Wikipedia(日本語版)](https://ja.wikipedia.org/wiki/ラゲールの陪多項式)の$k=0$の場合と見比べて頂きたい." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 66, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{0}(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{0}(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{1}(x) = 1.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{1}(x) = 1.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{2}(x) = 2.0 - 4.0 x + 1.0 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{2}(x) = 2.0 - 4.0 x + 1.0 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{3}(x) = 6.0 - 18.0 x + 9.0 x^{2} - 1.0 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{3}(x) = 6.0 - 18.0 x + 9.0 x^{2} - 1.0 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{4}(x) = 24.0 - 96.0 x + 72.0 x^{2} - 16.0 x^{3} + 1.0 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{4}(x) = 24.0 - 96.0 x + 72.0 x^{2} - 16.0 x^{3} + 1.0 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{5}(x) = 120.0 - 600.0 x + 600.0 x^{2} - 200.0 x^{3} + 25.0 x^{4} - 1.0 x^{5}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{5}(x) = 120.0 - 600.0 x + 600.0 x^{2} - 200.0 x^{3} + 25.0 x^{4} - 1.0 x^{5}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{6}(x) = 720.0 - 4320.0 x + 5400.0 x^{2} - 2400.0 x^{3} + 450.0 x^{4} - 36.0 x^{5} + 1.0 x^{6}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{6}(x) = 720.0 - 4320.0 x + 5400.0 x^{2} - 2400.0 x^{3} + 450.0 x^{4} - 36.0 x^{5} + 1.0 x^{6}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\tilde{L}_{7}(x) = 5040.0 - 35280.0 x + 52920.0 x^{2} - 29400.0 x^{3} + 7350.0 x^{4} - 882.0 x^{5} + 49.0 x^{6} - 1.0 x^{7}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$\\tilde{L}_{7}(x) = 5040.0 - 35280.0 x + 52920.0 x^{2} - 29400.0 x^{3} + 7350.0 x^{4} - 882.0 x^{5} + 49.0 x^{6} - 1.0 x^{7}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:7\n", | |
| " p = factorial(n)*basis(Laguerre{0}, n)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " latexstring(\"\\\\tilde{L}_{$n}(x) = \", latexify(\"$q\", env=:raw, cdot=false)) |> display\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## ラゲールの陪多項式\n", | |
| "\n", | |
| "[ラゲールの陪多項式](https://ja.wikipedia.org/wiki/ラゲールの陪多項式)$L_n^k(x)$は[水素原子のSchrödinger方程式の固有関数の動径部分](https://ja.wikipedia.org/wiki/水素原子におけるシュレーディンガー方程式の解#(W5)からのラゲールの陪多項式の導出)に登場する. これは定義ではないが\n", | |
| ", ラゲール多項式$L_n(x)$との関係\n", | |
| "\n", | |
| "$$\n", | |
| " L_n^{k}(x) = \\frac{\\mathrm{d}^k}{\\mathrm{d}x^k} L_n(x)\n", | |
| "$$\n", | |
| "\n", | |
| "が知られている. SpecialPolynomials.jlでは一般化ラゲール多項式$L_n^{(\\alpha)}(x)$が実装されているので, $L_n^{k}(x) = L_{n-k}^{(k)}(x)$という関係を利用すればよい. n^{k}(x)$は`basis(Laguerre{k}, n-k)(x)`である." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 67, | |
| "metadata": { | |
| "scrolled": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_0^0(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_0^0(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_1^0(x) = 1.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_1^0(x) = 1.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_1^1(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_1^1(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^0(x) = 1.0 - 2.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^0(x) = 1.0 - 2.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^1(x) = 2.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^1(x) = 2.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^2(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^2(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^0(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^0(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^1(x) = 3.0 - 3.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^1(x) = 3.0 - 3.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^2(x) = 3.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^2(x) = 3.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^3(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^3(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^0(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^0(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^1(x) = 4.0 - 6.0 x + 2.0 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^1(x) = 4.0 - 6.0 x + 2.0 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^2(x) = 6.0 - 4.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^2(x) = 6.0 - 4.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^3(x) = 4.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^3(x) = 4.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^4(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^4(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:4\n", | |
| " for k in 0:n\n", | |
| " p = basis(Laguerre{k}, n-k)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " latexstring(\"L_$n^$k(x) = \", latexify(\"$q\", env=:raw, cdot=false)) |> display\n", | |
| " end\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Polynomials.jlの`derivative(q,k)`を用いてラゲール多項式$L_n(x)$の$k$階の導関数を計算しても, 同様の結果が得られる." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 68, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_0^0(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_0^0(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_1^0(x) = 1.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_1^0(x) = 1.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_1^1(x) = -1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_1^1(x) = -1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^0(x) = 1.0 - 2.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^0(x) = 1.0 - 2.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^1(x) = -2.0 + 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^1(x) = -2.0 + 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^2(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^2(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^0(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^0(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^1(x) = -3.0 + 3.0 x - 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^1(x) = -3.0 + 3.0 x - 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^2(x) = 3.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^2(x) = 3.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^3(x) = -1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^3(x) = -1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^0(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^0(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^1(x) = -4.0 + 6.0 x - 2.0 x^{2} + 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^1(x) = -4.0 + 6.0 x - 2.0 x^{2} + 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^2(x) = 6.0 - 4.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^2(x) = 6.0 - 4.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^3(x) = -4.0 + 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^3(x) = -4.0 + 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^4(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^4(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:4\n", | |
| " for k in 0:n\n", | |
| " p = basis(Laguerre{0}, n)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " r = derivative(q,k)\n", | |
| " latexstring(\"L_$n^$k(x) = \", latexify(\"$r\", env=:raw, cdot=false)) |> display\n", | |
| " end\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 一般化ラゲール多項式\n", | |
| "\n", | |
| "一般化ラゲール多項式$L_n^{(\\alpha)}(x)$は[水素原子](https://en.wikipedia.org/wiki/Hydrogen-like_atom)や[モースポテンシャル](https://ja.wikipedia.org/wiki/モースポテンシャル)を考えるときに登場する.\n", | |
| "\n", | |
| "[Generalized Laguerre polynomials](https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials)の和訳. 少なくともGoogle翻訳ではラゲールの陪多項式と訳されるが, 日本で使われているラゲールの陪多項式とは定義が違うようなので, 注意されたい. 調査したところ, かなり紛らわしいので, 一般化ラゲール多項式$L_n^{(\\alpha)}(x)$とラゲールの陪多項式$L_n^k(x)$は別物と考えた方がすっきりする.「どうせ$k\\in\\mathbb{N}$と$k\\in\\mathbb{R}$の違いだろう」などと決めつけてかかると痛い目を見る.\n", | |
| "\n", | |
| "### 前提\n", | |
| "\n", | |
| "まず, 一般化ラゲール多項式$L_n^{(\\alpha)}(x)$とラゲールの陪多項式$L_n^k(x)$はそもそも満たすべき方程式が異なる.\n", | |
| "\n", | |
| "|多項式|満たすべき方程式|出典|\n", | |
| "|:---|:---|:---|\n", | |
| "|ラゲール多項式$L_n(x)$|$$xy'' + (1-x)y' + ny = 0$$||\n", | |
| "|ラゲールの陪多項式$L_n^k(x)$|$$xy'' + (k+1-x)y' + (n-k)y = 0$$|[日本語のWikipedia](https://ja.wikipedia.org/wiki/ラゲールの陪多項式)|\n", | |
| "|Generalized Laguerre polynomials<br>一般化ラゲール多項式$L_n^{(\\alpha)}(x)$|$$xy'' + (\\alpha+1-x)y' + ny = 0$$|[英語のWikipedia](https://en.wikipedia.org/wiki/Laguerre_polynomials)|\n", | |
| "\n", | |
| "### 共通点\n", | |
| "\n", | |
| "ラゲールの陪多項式$L_n^k(x)$も一般化ラゲール多項式$L_n^{(\\alpha)}(x)$も$k=\\alpha=0$のときにラゲール多項式$L_n(x)$に一致するように一般化されていることが明らかである.\n", | |
| "\n", | |
| "$$\n", | |
| " L_n^{0}(x) = L_n^{(0)}(x) = L_n(x)\n", | |
| "$$\n", | |
| "\n", | |
| "### 相違点\n", | |
| "\n", | |
| "一般化ラゲール多項式$L_n^{(\\alpha)}(x)$はラゲールの陪多項式$L_n^k(x)$とは異なり, $\\alpha\\in\\mathbb{N}$に限定したとしてもラゲール多項式$L_n(x)$の導関数ではない.\n", | |
| "\n", | |
| "$$\n", | |
| "\\begin{align}\n", | |
| " L_n^{k}(x) &= \\frac{\\mathrm{d}^k}{\\mathrm{d}x^k} L_n(x)\\\\\n", | |
| " L_n^{(\\alpha)}(x) &\\neq \\frac{\\mathrm{d}^\\alpha}{\\mathrm{d}x^\\alpha} L_n(x)\n", | |
| "\\end{align}\n", | |
| "$$\n", | |
| "\n", | |
| "これは, そもそも満たすべき方程式が違うためである. 逆に言うと, ラゲールの陪多項式$L_n^k(x)$もガンマ関数を用いて$k\\in\\mathbb{R}$で成り立つようにしてやれば(実際, ライブラリでは成り立つ), 満たすべき方程式以外に違いは無いともいえる. そうすると, 満たすべき方程式がどちらなのか確認さえすれば問題ない, ように考えがちだが, ラゲール多項式$L_n(x)$の$1/n!$の有無も文献によるので慎重になった方がよい.\n", | |
| "\n", | |
| "### 対応関係\n", | |
| "\n", | |
| "見ての通り, ラゲールの陪多項式$L_n^k(x)$と一般化ラゲール多項式$L_n^{(\\alpha)}(x)$は異なる性質を持つが, これらの間には次の関係が明らかに成り立つ.\n", | |
| "\n", | |
| "$$\n", | |
| " L_n^{k}(x) = L_{n-k}^{(k)}(x)\n", | |
| "$$\n", | |
| "\n", | |
| "このことから, 水素原子の動径関数は多くが$L_{2l+1}^{n+l}(x)$であるが, たまに$L_{2n+l}^{n-l-1}(x)$のパターンがある. 前者は[『マッカーリ・サイモン物理化学 上』p.224](http://www.tkd-pbl.com/book/b16539.html), [『アトキンス 物理化学（上）第10版』p.381](http://www.tkd-pbl.com/book/b279830.html), [原田義也『量子化学 上巻』(第2版, 2019, 裳華房) p.123](https://www.shokabo.co.jp/mybooks/ISBN978-4-7853-3073-6.htm), [高柳和夫『原子分子物理学』(2000, 朝倉書店) p.16](https://www.asakura.co.jp/detail.php?book_code=13681), [Wikipedia(日本語)](https://ja.wikipedia.org/wiki/%E6%B0%B4%E7%B4%A0%E5%8E%9F%E5%AD%90%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8B%E3%82%B7%E3%83%A5%E3%83%AC%E3%83%BC%E3%83%87%E3%82%A3%E3%83%B3%E3%82%AC%E3%83%BC%E6%96%B9%E7%A8%8B%E5%BC%8F%E3%81%AE%E8%A7%A3#(W5)%E3%81%8B%E3%82%89%E3%81%AE%E3%83%A9%E3%82%B2%E3%83%BC%E3%83%AB%E3%81%AE%E9%99%AA%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%81%AE%E5%B0%8E%E5%87%BA)などであり, 後者が[こちらのサイト](https://keisan.casio.jp/exec/system/1176445481)や[Wikipedia(英語)](https://en.wikipedia.org/wiki/Hydrogen-like_atom)である.\n", | |
| "\n", | |
| "この違いを抜きにしても規格化定数が間違っていたりするので, 数値積分などで簡単にでも確認した方がよい.\n", | |
| "\n", | |
| "### 使用法\n", | |
| "\n", | |
| "SpecialPolynomials.jlでサポートされており, $L_n^{(\\alpha)}(x)$はそのまま`basis(Laguerre{α}, n)(x)`である. もちろん$\\alpha\\in\\mathbb{R}$で使える." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 59, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_0^{(0)}(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_0^{(0)}(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_1^{(0)}(x) = 1.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_1^{(0)}(x) = 1.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_1^{(1)}(x) = 2.0 - 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_1^{(1)}(x) = 2.0 - 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^{(0)}(x) = 1.0 - 2.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^{(0)}(x) = 1.0 - 2.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^{(1)}(x) = 3.0 - 3.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^{(1)}(x) = 3.0 - 3.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_2^{(2)}(x) = 6.0 - 4.0 x + 0.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_2^{(2)}(x) = 6.0 - 4.0 x + 0.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^{(0)}(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^{(0)}(x) = 1.0 - 3.0 x + 1.5 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^{(1)}(x) = 4.0 - 6.0 x + 2.0 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^{(1)}(x) = 4.0 - 6.0 x + 2.0 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^{(2)}(x) = 10.0 - 10.0 x + 2.5 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^{(2)}(x) = 10.0 - 10.0 x + 2.5 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_3^{(3)}(x) = 20.0 - 15.0 x + 3.0 x^{2} - 0.166667 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_3^{(3)}(x) = 20.0 - 15.0 x + 3.0 x^{2} - 0.166667 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^{(0)}(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^{(0)}(x) = 1.0 - 4.0 x + 3.0 x^{2} - 0.666667 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^{(1)}(x) = 5.0 - 10.0 x + 5.0 x^{2} - 0.833333 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^{(1)}(x) = 5.0 - 10.0 x + 5.0 x^{2} - 0.833333 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^{(2)}(x) = 15.0 - 20.0 x + 7.5 x^{2} - 1.0 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^{(2)}(x) = 15.0 - 20.0 x + 7.5 x^{2} - 1.0 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^{(3)}(x) = 35.0 - 35.0 x + 10.5 x^{2} - 1.16667 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^{(3)}(x) = 35.0 - 35.0 x + 10.5 x^{2} - 1.16667 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$L_4^{(4)}(x) = 70.0 - 56.0 x + 14.0 x^{2} - 1.33333 x^{3} + 0.0416667 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$L_4^{(4)}(x) = 70.0 - 56.0 x + 14.0 x^{2} - 1.33333 x^{3} + 0.0416667 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:4\n", | |
| " for k in 0:n\n", | |
| " p = basis(Laguerre{k}, n)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " latexstring(\"L_$n^{($k)}(x) = \", latexify(\"$q\", env=:raw, cdot=false)) |> display\n", | |
| " end\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## ルジャンドル多項式\n", | |
| "\n", | |
| "[ルジャンドル多項式](https://ja.wikipedia.org/wiki/%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E5%A4%9A%E9%A0%85%E5%BC%8F)$P_n$はロドリゲスの公式\n", | |
| "\n", | |
| "$$\n", | |
| " P_n(x) = \\frac{1}{2^n n!} \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left[ \\left( x^2-1 \\right)^2 \\right]\n", | |
| "$$\n", | |
| "\n", | |
| "で与えられる. $P_n(x)$は`basis(Legendre, n)(x)`のように書けばよい. \n", | |
| "[こちら](https://ja.wikipedia.org/wiki/%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E5%A4%9A%E9%A0%85%E5%BC%8F#%E5%B8%B0%E7%B4%8D%E7%9A%84%E5%AE%9A%E7%BE%A9)と見比べて頂きたい." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 75, | |
| "metadata": { | |
| "scrolled": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{0}(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{0}(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{1}(x) = 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{1}(x) = 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{2}(x) = -0.5 + 1.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{2}(x) = -0.5 + 1.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{3}(x) = -1.5 x + 2.5 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{3}(x) = -1.5 x + 2.5 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{4}(x) = 0.375 - 3.75 x^{2} + 4.375 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{4}(x) = 0.375 - 3.75 x^{2} + 4.375 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{5}(x) = 1.875 x - 8.75 x^{3} + 7.875 x^{5}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{5}(x) = 1.875 x - 8.75 x^{3} + 7.875 x^{5}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{6}(x) = -0.3125 + 6.5625 x^{2} - 19.6875 x^{4} + 14.4375 x^{6}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{6}(x) = -0.3125 + 6.5625 x^{2} - 19.6875 x^{4} + 14.4375 x^{6}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{7}(x) = -2.1875 x + 19.6875 x^{3} - 43.3125 x^{5} + 26.8125 x^{7}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{7}(x) = -2.1875 x + 19.6875 x^{3} - 43.3125 x^{5} + 26.8125 x^{7}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{8}(x) = 0.273438 - 9.84375 x^{2} + 54.1406 x^{4} - 93.8438 x^{6} + 50.2734 x^{8}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{8}(x) = 0.273438 - 9.84375 x^{2} + 54.1406 x^{4} - 93.8438 x^{6} + 50.2734 x^{8}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{9}(x) = 2.46094 x - 36.0938 x^{3} + 140.766 x^{5} - 201.094 x^{7} + 94.9609 x^{9}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{9}(x) = 2.46094 x - 36.0938 x^{3} + 140.766 x^{5} - 201.094 x^{7} + 94.9609 x^{9}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{10}(x) = -0.246094 + 13.5352 x^{2} - 117.305 x^{4} + 351.914 x^{6} - 427.324 x^{8} + 180.426 x^{10}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{10}(x) = -0.246094 + 13.5352 x^{2} - 117.305 x^{4} + 351.914 x^{6} - 427.324 x^{8} + 180.426 x^{10}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{11}(x) = -2.70703 x + 58.6523 x^{3} - 351.914 x^{5} + 854.648 x^{7} - 902.129 x^{9} + 344.449 x^{11}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{11}(x) = -2.70703 x + 58.6523 x^{3} - 351.914 x^{5} + 854.648 x^{7} - 902.129 x^{9} + 344.449 x^{11}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for n in 0:11\n", | |
| " p = basis(Legendre, n)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " latexstring(\"P_{$n}(x) = \", latexify(\"$q\", env=:raw, cdot=false)) |> display\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## ルジャンドルの陪多項式\n", | |
| "\n", | |
| "[ルジャンドルの陪多項式](https://ja.wikipedia.org/wiki/%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E5%A4%9A%E9%A0%85%E5%BC%8F#%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E9%99%AA%E5%A4%9A%E9%A0%85%E5%BC%8F)$P_k^m(t)$は[球面調和関数](https://ja.wikipedia.org/wiki/%E7%90%83%E9%9D%A2%E8%AA%BF%E5%92%8C%E9%96%A2%E6%95%B0#3%E6%AC%A1%E5%85%83%E7%A9%BA%E9%96%93%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8B%E7%90%83%E9%9D%A2%E8%AA%BF%E5%92%8C%E9%96%A2%E6%95%B0)すなわち水素原子の軌道の角度部分に登場する. ルジャンドル多項式$P_n$との関係が次式で与えられている.\n", | |
| "\n", | |
| "$$\n", | |
| " P_k^m(t) = \\left( 1-t^2 \\right)^{m/2} \\frac{\\mathrm{d}^m P_k(t)}{\\mathrm{d}t^m}\n", | |
| "$$\n", | |
| "\n", | |
| "なお, [Wikipediaの英語版](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials)だと符号がつけ加えられているが, 結局は線形微分方程式の解なので定数倍の違いは著者の好みに委ねられてしまう. 定義が違えば球面調和関数の規格化定数も変わるので注意されたい.\n", | |
| "\n", | |
| "$$\n", | |
| " P_k^m(t) = (-1)^m \\left( 1-t^2 \\right)^{m/2} \\frac{\\mathrm{d}^m P_k(t)}{\\mathrm{d}t^m}\n", | |
| "$$\n", | |
| "\n", | |
| "ここでは英語版の定義を採用する. [こちら](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials#The_first_few_associated_Legendre_functions)と比較して頂きたい." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 123, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{0}^{0}(x) = 1.0$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{0}^{0}(x) = 1.0$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{1}^{0}(x) = 1.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{1}^{0}(x) = 1.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{1}^{1}(x) = \\left(-1.0\\right)\\sqrt{1-x^2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{1}^{1}(x) = \\left(-1.0\\right)\\sqrt{1-x^2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{2}^{0}(x) = -0.5 + 1.5 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{2}^{0}(x) = -0.5 + 1.5 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{2}^{1}(x) = \\left(-3.0 x\\right)\\sqrt{1-x^2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{2}^{1}(x) = \\left(-3.0 x\\right)\\sqrt{1-x^2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{2}^{2}(x) = 3.0 - 6.0 x$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{2}^{2}(x) = 3.0 - 6.0 x$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{3}^{0}(x) = -1.5 x + 2.5 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{3}^{0}(x) = -1.5 x + 2.5 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{3}^{1}(x) = \\left(1.5 - 7.5 x^{2}\\right)\\sqrt{1-x^2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{3}^{1}(x) = \\left(1.5 - 7.5 x^{2}\\right)\\sqrt{1-x^2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{3}^{2}(x) = 15.0 x - 30.0 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{3}^{2}(x) = 15.0 x - 30.0 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{3}^{3}(x) = \\left(-15.0 + 30.0 x\\right)\\sqrt{1-x^2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{3}^{3}(x) = \\left(-15.0 + 30.0 x\\right)\\sqrt{1-x^2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{4}^{0}(x) = 0.375 - 3.75 x^{2} + 4.375 x^{4}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{4}^{0}(x) = 0.375 - 3.75 x^{2} + 4.375 x^{4}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{4}^{1}(x) = \\left(7.5 x - 17.5 x^{3}\\right)\\sqrt{1-x^2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{4}^{1}(x) = \\left(7.5 x - 17.5 x^{3}\\right)\\sqrt{1-x^2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{4}^{2}(x) = -7.5 + 15.0 x + 52.5 x^{2} - 105.0 x^{3}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{4}^{2}(x) = -7.5 + 15.0 x + 52.5 x^{2} - 105.0 x^{3}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{4}^{3}(x) = \\left(-105.0 x + 210.0 x^{2}\\right)\\sqrt{1-x^2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{4}^{3}(x) = \\left(-105.0 x + 210.0 x^{2}\\right)\\sqrt{1-x^2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| }, | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$P_{4}^{4}(x) = 105.0 - 420.0 x + 420.0 x^{2}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$P_{4}^{4}(x) = 105.0 - 420.0 x + 420.0 x^{2}$\"" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "# using Latexify\n", | |
| "# using LaTeXStrings\n", | |
| "\n", | |
| "for k in 0:4\n", | |
| " for m in 0:k\n", | |
| " p = basis(Legendre, k)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " r = derivative(q, abs(m))\n", | |
| " if m % 2 == 0\n", | |
| " s = (-1)^m*(Polynomial([1,-2]))^(Int(m/2))*r\n", | |
| " latexstring(\"P_{$k}^{$m}(x) = \", latexify(\"$s\", env=:raw, cdot=false)) |> display\n", | |
| " else\n", | |
| " s = (-1)^m*(Polynomial([1,-2]))^(Int((m-1)/2))*r\n", | |
| " latexstring(\"P_{$k}^{$m}(x) = \\\\left(\", latexify(\"$s\", env=:raw, cdot=false), \"\\\\right)\\\\sqrt{1-x^2}\") |> display\n", | |
| " end\n", | |
| " end\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "関数として利用する場合は次のようにする." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 140, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "P (generic function with 1 method)" | |
| ] | |
| }, | |
| "execution_count": 140, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialPolynomials\n", | |
| "# using Polynomials\n", | |
| "\n", | |
| "function P(k,m,x)\n", | |
| " if m<0\n", | |
| " return (-1)^m * factorial(l-m) / factorial(l+m) * P(k,m,x)\n", | |
| " end\n", | |
| " p = basis(Legendre, k)\n", | |
| " q = convert(Polynomial, p)\n", | |
| " r = derivative(q, abs(m))\n", | |
| " return (-1)^m * (1-x^2)^(m/2) * r(x)\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 155, | |
| "metadata": { | |
| "scrolled": false | |
| }, | |
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| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# Plots\n", | |
| "\n", | |
| "for m in 0:2\n", | |
| " plot(title=latexstring(\"m=$m\"), xlabel=L\"x\", ylabel=latexstring(\"P_k^m(x)\"), xlims=(-1,1), ylims=(-[1.2,3,15][m+1],[1.2,3,23][m+1]), legend=:bottomright)\n", | |
| " for k in 0:[4,5,6][m+1]\n", | |
| " plot!(x->P(k,m,x), label=latexstring(\"k=$k\"), lw=2)\n", | |
| " end\n", | |
| " plot!() |> display\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## ガンマ関数\n", | |
| "\n", | |
| "ガンマ関数$\\Gamma(x)$は階乗の一般化であり, $\\Gamma(n) = (n-1)!$という関係が成立する. まずこれを確認する. Juliaでは階乗は`factorial(x)`で利用できる. ガンマ関数はSpecialFunctions.jlでサポートされており, `gamma(x)`である." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 58, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "n\t(n-1)!\tΓ(n)\n", | |
| "1\t1\t1.0\n", | |
| "2\t1\t1.0\n", | |
| "3\t2\t2.0\n", | |
| "4\t6\t6.0\n", | |
| "5\t24\t24.0\n", | |
| "6\t120\t120.0\n", | |
| "7\t720\t720.0\n", | |
| "8\t5040\t5040.0\n", | |
| "9\t40320\t40320.0\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialFunctions\n", | |
| "println(\"n\\t(n-1)!\\tΓ(n)\")\n", | |
| "\n", | |
| "for i in 1:9\n", | |
| " x = factorial(i-1)\n", | |
| " y = gamma(i)\n", | |
| " println(\"$i\\t$x\\t$y\")\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "非整数についても$\\Gamma(1/2) = \\sqrt{\\pi}$などが成り立つことが確認できる." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 158, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "3.1415926535897936" | |
| ] | |
| }, | |
| "execution_count": 158, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialFunctions\n", | |
| "gamma(1/2)^2" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 誤差関数\n", | |
| "\n", | |
| "[誤差関数](https://ja.wikipedia.org/wiki/%E8%AA%A4%E5%B7%AE%E9%96%A2%E6%95%B0)はHartree-Fock法の核-電子間引力の積分などに登場する. SpecialFunctions.jlでサポートされており, 誤差関数は`erf(x)`, 相補誤算関数は`erfc(x)`である. たまにお世話になるので一応, 簡単な数値積分を用いて検証しておく.\n", | |
| "\n", | |
| "$$\n", | |
| " \\mathrm{erf}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_0^x \\exp(-t^2) \\mathrm{d}t\\\\\n", | |
| " \\mathrm{erfc}(x) = 1 - \\mathrm{erf}(x)\n", | |
| "$$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 261, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "integral (generic function with 1 method)" | |
| ] | |
| }, | |
| "execution_count": 261, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# 数値積分のルーチン\n", | |
| "function integral(func, x_min, x_max, dx)\n", | |
| " sum = 0.0\n", | |
| " for x in x_min:dx:x_max\n", | |
| " sum += func(x)*dx\n", | |
| " end\n", | |
| " return sum\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 262, | |
| "metadata": { | |
| "scrolled": true | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| " x \t erf(x) \t 1-erfc(x)\t integral\n", | |
| " 0.000000\t 0.000000\t 0.000000\t 0.000000\n", | |
| " 0.500000\t 0.520500\t 0.520500\t 0.520500\n", | |
| " 1.000000\t 0.842701\t 0.842701\t 0.842701\n", | |
| " 1.500000\t 0.966105\t 0.966105\t 0.966105\n", | |
| " 2.000000\t 0.995322\t 0.995322\t 0.995322\n", | |
| " 2.500000\t 0.999593\t 0.999593\t 0.999593\n", | |
| " 3.000000\t 0.999978\t 0.999978\t 0.999978\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "# using SpecialFunctions\n", | |
| "# using Printf\n", | |
| "\n", | |
| "@printf(\"%-9s\\t%-9s\\t%-9s\\t%-9s\\n\", \" x\", \" erf(x)\", \" 1-erfc(x)\", \" integral\")\n", | |
| "for x in 0:0.5:3\n", | |
| " @printf(\"%9f\\t%9f\\t%9f\\t%9f\\n\", x, erf(x), 1-erfc(x), 2/sqrt(pi)*integral(t->exp(-t^2), 0, x, 0.0000001))\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 動作環境" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 62, | |
| "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()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 参考文献\n", | |
| "\n", | |
| "- [SpecialFunctions.jl](https://specialfunctions.juliamath.org/stable/functions_overview/)\n", | |
| "- [SpecialPolynomials.jl](https://docs.juliahub.com/SpecialPolynomials/)\n", | |
| "- [Polynomials.jl](https://juliamath.github.io/Polynomials.jl/stable/), [微分演算](https://juliamath.github.io/Polynomials.jl/stable/reference/#Polynomials.derivative), [積分演算](https://juliamath.github.io/Polynomials.jl/stable/reference/#Polynomials.integrate)\n", | |
| "- [Latexify.jl](https://korsbo.github.io/Latexify.jl/stable/)\n", | |
| "- [LaTeXStrings](https://github.com/stevengj/LaTeXStrings.jl)\n", | |
| "- [ガンマ関数](https://ja.wikipedia.org/wiki/%E3%82%AC%E3%83%B3%E3%83%9E%E9%96%A2%E6%95%B0)\n", | |
| "- [ラゲールの陪多項式](https://ja.wikipedia.org/wiki/ラゲールの陪多項式)\n", | |
| "- [水素原子のSchrödinger方程式の固有関数の動径部分](https://ja.wikipedia.org/wiki/水素原子におけるシュレーディンガー方程式の解#(W5)からのラゲールの陪多項式の導出)\n", | |
| "- [Generalized Laguerre polynomials](https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials)\n", | |
| "- [ルジャンドル多項式](https://ja.wikipedia.org/wiki/%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E5%A4%9A%E9%A0%85%E5%BC%8F#%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E9%99%AA%E5%A4%9A%E9%A0%85%E5%BC%8F)\n", | |
| "- [ルジャンドルの陪多項式](https://ja.wikipedia.org/wiki/%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E5%A4%9A%E9%A0%85%E5%BC%8F#%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB%E9%99%AA%E5%A4%9A%E9%A0%85%E5%BC%8F)\n", | |
| "- [球面調和関数](https://ja.wikipedia.org/wiki/%E7%90%83%E9%9D%A2%E8%AA%BF%E5%92%8C%E9%96%A2%E6%95%B0#3%E6%AC%A1%E5%85%83%E7%A9%BA%E9%96%93%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8B%E7%90%83%E9%9D%A2%E8%AA%BF%E5%92%8C%E9%96%A2%E6%95%B0)\n", | |
| "\n", | |
| "Plots.jlの使い方についてはこちらの記事を参照されたい.\n", | |
| "https://zenn.dev/ohno/articles/3101433fbe9231\n", | |
| "\n", | |
| "この記事の元になったJupyter Notebookのデータは下記のリンクにある.\n", | |
| "https://gist.github.com/ohno/1ed4da972acbc988bc318fc53647afac" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Julia 1.6.2", | |
| "language": "julia", | |
| "name": "julia-1.6" | |
| }, | |
| "language_info": { | |
| "file_extension": ".jl", | |
| "mimetype": "application/julia", | |
| "name": "julia", | |
| "version": "1.6.2" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 4 | |
| } |
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