2020年1月16日木曜日

学習環境

解析入門(上) (松坂和夫 数学入門シリーズ 4) (松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題7の解答を求めてみる。



    1. x 2 n + 1 e - x 2 dx = x 2 n + 1 2 n + 1 e - x 2 - 1 2 n + 1 x 2 n + 1 e - x 2 - 2 x dx = x 2 n + 1 2 n + 1 e - x 2 + 1 n + 1 x 2 n + 1 + 1 e - x 2 dx 0 x 2 n + 1 e - x 2 dx = 1 n + 1 0 x 2 n + 1 + 1 e - x 2 dx 0 x 2 n + 1 + 1 e - x 2 dx = n + 1 0 x 2 n + 1 e - x 2 dx 0 x e - x 2 dx = lim b 0 b x e - x 2 dx = - 1 2 lim b e - x 2 0 b = - 1 2 lim b e - b 2 - 1 = 1 2

      よって、

      0 n x 2 n + 1 e - x 2 dx = n ! 2

    2. x α log x n dx = x α + 1 α + 1 log x n - 1 α + 1 x α + 1 n log x n - 1 · 1 x dx = x α + 1 α + 1 log x n - n α + 1 x α log x n - 1 dx 0 1 x α log x n dx = - n α + 1 0 1 x α log x n - 1 dx 0 1 x α log x dx = - 1 α + 1 0 1 x α dx = - 1 α + 1 2 x α + 1 0 1 = - 1 α + 1 2

      よって、

      0 1 x α log x n dx = - 1 n n ! x + 1 2 + n - 1 = - 1 n n ! α + 1 n + 1

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import pprint, symbols, Integral, log, exp, oo, Rational, plot

print('7.')

x = symbols('x', real=True)
n = symbols('n', integer=True, nonnegative=True)
f = x ** (2 * n + 1) * exp(-x ** 2)
a = 2
g1 = x ** -Rational(1, 2) * log(x) ** n
g2 = x ** 2 * log(x) ** n
xs = [(0, oo), (0, 1), (0, 1)]

for i, (h, (x1, x2)) in enumerate(zip([f, g1, g2], xs), 1):
    print(f'({i})')
    I = Integral(h, (x, x1, x2))
    for o in [I, I.doit()]:
        pprint(o.simplify())
        print()


p = plot(*[h.subs({n: n0})
           for h in [f, g1, g2]
           for n0 in range(3)],
         (x, -5, 5),
         ylim=(-5, 5),
         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(p, colors):
    pprint(o)
p.show()
p.save('sample7.png')

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample7.py  
7.
(1)
∞                 
⌠                 
⎮             2   
⎮  2⋅n + 1  -x    
⎮ x       ⋅ℯ    dx
⌡                 
0                 

n!
──
2 

(2)
1           
⌠           
⎮    n      
⎮ log (x)   
⎮ ─────── dx
⎮    √x     
⌡           
0           

1           
⌠           
⎮    n      
⎮ log (x)   
⎮ ─────── dx
⎮    √x     
⌡           
0           

(3)
1              
⌠              
⎮  2    n      
⎮ x ⋅log (x) dx
⌡              
0              

1              
⌠              
⎮  2    n      
⎮ x ⋅log (x) dx
⌡              
0              

(cartesian line: x*exp(-x**2) for x over (-5.0, 5.0), red)
(cartesian line: x**3*exp(-x**2) for x over (-5.0, 5.0), green)
(cartesian line: x**5*exp(-x**2) for x over (-5.0, 5.0), blue)
(cartesian line: 1/sqrt(x) for x over (-5.0, 5.0), brown)
(cartesian line: log(x)/sqrt(x) for x over (-5.0, 5.0), orange)
(cartesian line: log(x)**2/sqrt(x) for x over (-5.0, 5.0), purple)
(cartesian line: x**2 for x over (-5.0, 5.0), pink)
(cartesian line: x**2*log(x) for x over (-5.0, 5.0), gray)
(cartesian line: x**2*log(x)**2 for x over (-5.0, 5.0), skyblue)
%

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