数学 - Python - 解析学 - 級数 - べき級数の微分と積分 - 項別微分、等式の証明、第1種ベッセル関数(besselj)

解析入門 原書第3版
原著
楽天ブックス Yahoo!

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• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス Yahoo!)
  • JavaScriptリファレンス 第6版 (楽天ブックス(Kobo) Yahoo!)
  • インタラクティブ・データビジュアライゼーション (楽天ブックス(Kobo) Yahoo!)

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第4部(級数)、第15章(級数)、7(べき級数の微分と積分)の練習問題5を求めてみる。

5. 
   
   $\begin{array}{l}J\left(x\right)\\ =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{{\left(n!\right)}^2}{\left(\frac{x}{2}\right)}^{2n}\\ =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{{\left(n!\right)}^2{2}^{2n}}{x}^{2n}\\ \frac{d}{\mathrm{dx}}J\left(x\right)\\ =\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{{\left(n!\right)}^2{2}^{2n}}·2n{x}^{2n-1}\\ =\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}n}{{\left(n!\right)}^2{2}^{2n-1}}{x}^{2n-1}\\ =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n+1}\left(n+1\right)}{{\left(\left(n+1\right)!\right)}^2{2}^{2n+1}}{x}^{2n+1}\\ \frac{d^2}{d{x}^2}J\left(x\right)\\ =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n+1}\left(n+1\right)}{{\left(\left(n+1\right)!\right)}^2{2}^{2n+1}}\left(2n+1\right)x^{2n}\end{array}$

   よって、

   $\begin{array}{l}{x}^{2}J\text{'}\text{'}\left(x\right)+xJ\text{'}\left(x\right)+x^{2}J\left(x\right)\\ =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n+1}\left(n+1\right)}{{\left(\left(n+1\right)!\right)}^2{2}^{2n+1}}\left(2n+1\right)x^{2n+2}\\ +\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n+1}\left(n+1\right)}{{\left(\left(n+1\right)!\right)}^2{2}^{2n+1}}{x}^{2n+2}\\ +\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{{\left(n!\right)}^2{2}^{2n}}{x}^{2n+2}\\ =\sum _{n=0}^{\infty }\frac{-{\left(-1\right)}^n\left(n+1\right)\left(2n+1\right)-{\left(-1\right)}^n\left(n+1\right)+{\left(-1\right)}^n{\left(n+1\right)}^{2}2}{{\left(\left(n+1\right)!\right)}^2{2}^{2n+1}}{x}^{2n+2}\\ =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n\left(-2n^2-3n-1-n-1+2n^2+4n+2\right)}{{\left(\left(n+1\right)!\right)}^2{2}^{2n+1}}{x}^{2n+2}\\ =0\end{array}$

   （証明終）

コード

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, summation, oo, plot, Derivative, factorial

print('5.')

x, n = symbols('x, n')
t = (-1) ** n / factorial(n) ** 2 * (x / 2) ** (2 * n)
f = summation(t, (n, 0, oo))
f1 = Derivative(f, x, 1).doit()
f2 = Derivative(f, x, 2).doit()
for o in [f, f1, f2, (x ** 2 * f2 + x * f1 + x ** 2 * f).simplify()]:
    pprint(o)
    print()


def g(m):
    return sum([t.subs({n: k}) for k in range(m + 1)])


fs = [1 / (1 + x ** 2)] + [g(m) for m in range(9)]
p = plot(*fs,
         (x, -2, 2),
         ylim=(-2, 2),
         legend=False,
         show=False)
colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']

for o, color in zip(p, colors):
    o.line_color = color

for o in zip(fs, colors):
    pprint(o)
    print()

p.show()
p.save('sample5.png')

入出力結果(Zsh、cmd.exe(コマンドプロンプト)、Terminal、Jupyter(IPython))

% ./sample5.py
5.
besselj(0, x)

-besselj(1, x)

-besselj(0, x) + besselj(2, x)
──────────────────────────────
              2               

0

⎛  1        ⎞
⎜──────, red⎟
⎜ 2         ⎟
⎝x  + 1     ⎠

(1, green)

⎛     2      ⎞
⎜    x       ⎟
⎜1 - ──, blue⎟
⎝    4       ⎠

⎛ 4    2           ⎞
⎜x    x            ⎟
⎜── - ── + 1, brown⎟
⎝64   4            ⎠

⎛    6     4    2            ⎞
⎜   x     x    x             ⎟
⎜- ──── + ── - ── + 1, orange⎟
⎝  2304   64   4             ⎠

⎛   8       6     4    2            ⎞
⎜  x       x     x    x             ⎟
⎜────── - ──── + ── - ── + 1, purple⎟
⎝147456   2304   64   4             ⎠

⎛     10         8       6     4    2          ⎞
⎜    x          x       x     x    x           ⎟
⎜- ──────── + ────── - ──── + ── - ── + 1, pink⎟
⎝  14745600   147456   2304   64   4           ⎠

⎛    12          10         8       6     4    2          ⎞
⎜   x           x          x       x     x    x           ⎟
⎜────────── - ──────── + ────── - ──── + ── - ── + 1, gray⎟
⎝2123366400   14745600   147456   2304   64   4           ⎠

⎛       14            12          10         8       6     4    2             
⎜      x             x           x          x       x     x    x              
⎜- ──────────── + ────────── - ──────── + ────── - ──── + ── - ── + 1, skyblue
⎝  416179814400   2123366400   14745600   147456   2304   64   4              

⎞
⎟
⎟
⎠

⎛       16              14            12          10         8       6     4  
⎜      x               x             x           x          x       x     x   
⎜─────────────── - ──────────── + ────────── - ──────── + ────── - ──── + ── -
⎝106542032486400   416179814400   2123366400   14745600   147456   2304   64  

  2            ⎞
 x             ⎟
 ── + 1, yellow⎟
 4             ⎠

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