数学 - Python - 解析学 - 各種の初等関数 - 三角関数(続き)、逆三角関数(高階微分、帰納法)

解析入門〈1〉数/各種の初等関数/各種の初等関数/各種の初等関数/各種の初等関数 (岩波オンデマンドブックス) オンデマンド (ペーパーバック) – 2016/7/12

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
Pythonからはじめる数学入門(参考書籍)

解析入門〈1〉(松坂和夫(著)、岩波書店)の第5章(各種の初等関数)、5.4(三角関数(続き)、逆三角関数)、問題4、5.を取り組んでみる。

$\begin{array}{l} n = 1 \\ \frac{d}{d x} (e^{x} \sin x) = e^{x} \sin x + e^{x} \cos x \\ \sqrt{2} e^{x} \sin (x + \frac{π}{4}) \\ = \sqrt{2} e^{x} (\sin x \cos \frac{π}{4} + \cos x \sin \frac{π}{4}) \\ = \sqrt{2} e^{x} (\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x) \\ = e^{x} \sin x + e^{x} \cos x \\ = e^{x} \sqrt{2} (\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x) \\ = e^{x} \sqrt{2} (\sin x \cos \frac{π}{4} + \cos x \sin \frac{π}{4}) \\ = \sqrt{2} e^{x} \sin (x + \frac{π}{4}) \\ \frac{d}{d x} ({(\sqrt{2})}^{n} e^{x} \sin (x + \frac{n π}{4})) \\ = {(\sqrt{2})}^{n} (e^{x} \sin (x + \frac{n π}{4}) + e^{x} \cos (x + \frac{n π}{4})) \\ = {(\sqrt{2})}^{n + 1} e^{x} (\sin (x + \frac{n π}{4}) \frac{1}{\sqrt{2}} + \cos (x + \frac{n π}{4}) \frac{1}{\sqrt{2}}) \\ = {(\sqrt{2})}^{n + 1} e^{x} (\sin (x + \frac{n π}{4}) \cos \frac{π}{4} + \cos (x + \frac{n π}{4}) \sin \frac{π}{4}) \\ = {(\sqrt{2})}^{n + 1} e^{x} \sin (x + \frac{n π}{4} + \frac{π}{4}) \\ = {(\sqrt{2})}^{n + 1} e^{x} \sin (x + \frac{(n + 1) π}{4}) \end{array}$
1. $\begin{array}{l} f' (x) = - a \frac{1}{x} \sin (\log x) + b \frac{1}{x} \cos (\log x) \\ f'' (x) = - a (- \frac{1}{x^{2}} \sin (\log x) + \frac{1}{x^{2}} \cos (\log x)) + b (- \frac{1}{x^{2}} \cos (\log x) - \frac{1}{x^{2}} \sin (\log x)) \\ - a (\cos (\log x) - \sin (\log x)) + b (- \cos (\log x) - \sin (\log x)) + \\ - a \sin (\log x) + b \cos (\log x) + \\ a \cos (\log x) + b \sin (\log x) \\ = 0 \end{array}$
2. $\begin{array}{l} n = 1 \\ \frac{d}{d x} x (a \cos (\log x) + b \sin (\log x)) \\ = (a \cos (\log x) + b \sin (\log x)) + x (- a \frac{1}{x} \sin (\log x) + b \frac{1}{x} \cos (\log x)) \\ = (a + b) \cos (\log x) + (b - a) \sin (\log x) \\ a_{1} = a + b, a_{2} = b - a \\ x^{n + 1} f^{(n + 1)} (x) \\ = x^{n + 1} \frac{d}{d x} f^{(n)} (x) \\ = x^{n + 1} \frac{d}{d x} (\frac{a_{n} \cos (\log x) + b_{n} \sin (\log x)}{x^{n}}) \\ = x^{n + 1} (\frac{(- a_{n} \frac{1}{x} \sin (\log x) + b_{n} \frac{1}{x} \cos (\log x)) x^{n} - (a_{n} \cos (\log x) + b_{n} \sin (\log x)) n x^{n - 1}}{x^{2 n}}) \\ = \frac{x^{n - 1} ((b_{n} - a_{n}) \cos (\log x) + (- a_{n} - b_{n}) \sin (\log x))}{x^{n - 1}} \\ = (b_{n} - a_{n}) \cos (\log x) + (- a_{n} - b_{n}) \sin (\log x) \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import Symbol, symbols, Derivative, sqrt, sin, cos, exp, pi, log, solve, pprint

print('4')
n = Symbol('n', integer=True, positive=True)
x = Symbol('x')

expr1 = exp(x) * sin(x)
expr2 = sqrt(2) ** n * exp(x) * sin(x + n * pi / 4)
pprint(expr1)
pprint(expr2)

for i in range(10):
    print('n = {}'.format(i + 1))
    d = Derivative(expr1, x, i).doit()
    pprint(d)
    expr = expr2.subs({n: i})
    pprint(expr)
    print((d - expr).expand() == 0)

print('5.')
a, b = symbols('a b')
x = Symbol('x', nonzero=True)
f = a * cos(log(x)) + b * sin(log(x))
pprint(f)

print('(a)')
f1 = Derivative(f, x, 1).doit()
f2 = Derivative(f, x, 2).doit()
pprint(f1)
pprint(f2)

expr = x ** 2 * f2 + x * f1 + f
pprint(expr)
print(expr.expand() == 0)

print('(b)')
for i in range(10):
    print('n = {0}'.format(i + 1))
    expr = x**n * Derivative(f, x, i).doit()
    pprint(expr)

入出力結果(Terminal, IPython)

$ ./sample4.py
4
 x       
ℯ ⋅sin(x)
 n                
 ─                
 2  x    ⎛π⋅n    ⎞
2 ⋅ℯ ⋅sin⎜─── + x⎟
         ⎝ 4     ⎠
n = 1
 x       
ℯ ⋅sin(x)
 x       
ℯ ⋅sin(x)
True
n = 2
 x           x       
ℯ ⋅sin(x) + ℯ ⋅cos(x)
    x    ⎛    π⎞
√2⋅ℯ ⋅sin⎜x + ─⎟
         ⎝    4⎠
False
n = 3
   x       
2⋅ℯ ⋅cos(x)
   x       
2⋅ℯ ⋅cos(x)
True
n = 4
                      x
2⋅(-sin(x) + cos(x))⋅ℯ 
      x    ⎛    π⎞
2⋅√2⋅ℯ ⋅cos⎜x + ─⎟
           ⎝    4⎠
False
n = 5
    x       
-4⋅ℯ ⋅sin(x)
    x       
-4⋅ℯ ⋅sin(x)
True
n = 6
                      x
-4⋅(sin(x) + cos(x))⋅ℯ 
       x    ⎛    π⎞
-4⋅√2⋅ℯ ⋅sin⎜x + ─⎟
            ⎝    4⎠
False
n = 7
    x       
-8⋅ℯ ⋅cos(x)
    x       
-8⋅ℯ ⋅cos(x)
True
n = 8
                     x
8⋅(sin(x) - cos(x))⋅ℯ 
       x    ⎛    π⎞
-8⋅√2⋅ℯ ⋅cos⎜x + ─⎟
            ⎝    4⎠
False
n = 9
    x       
16⋅ℯ ⋅sin(x)
    x       
16⋅ℯ ⋅sin(x)
True
n = 10
                      x
16⋅(sin(x) + cos(x))⋅ℯ 
       x    ⎛    π⎞
16⋅√2⋅ℯ ⋅sin⎜x + ─⎟
            ⎝    4⎠
False
5.
a⋅cos(log(x)) + b⋅sin(log(x))
(a)
  a⋅sin(log(x))   b⋅cos(log(x))
- ───────────── + ─────────────
        x               x      
a⋅sin(log(x)) - a⋅cos(log(x)) - b⋅sin(log(x)) - b⋅cos(log(x))
─────────────────────────────────────────────────────────────
                               2                             
                              x                              
                                  ⎛  a⋅sin(log(x))   b⋅cos(log(x))⎞
a⋅sin(log(x)) - b⋅cos(log(x)) + x⋅⎜- ───────────── + ─────────────⎟
                                  ⎝        x               x      ⎠
True
(b)
n = 1
 n                                
x ⋅(a⋅cos(log(x)) + b⋅sin(log(x)))
n = 2
 n ⎛  a⋅sin(log(x))   b⋅cos(log(x))⎞
x ⋅⎜- ───────────── + ─────────────⎟
   ⎝        x               x      ⎠
n = 3
 n                                                                
x ⋅(a⋅sin(log(x)) - a⋅cos(log(x)) - b⋅sin(log(x)) - b⋅cos(log(x)))
──────────────────────────────────────────────────────────────────
                                 2                                
                                x                                 
n = 4
 n                                                                     
x ⋅(-a⋅sin(log(x)) + 3⋅a⋅cos(log(x)) + 3⋅b⋅sin(log(x)) + b⋅cos(log(x)))
───────────────────────────────────────────────────────────────────────
                                    3                                  
                                   x                                   
n = 5
     n                                 
-10⋅x ⋅(a⋅cos(log(x)) + b⋅sin(log(x))) 
───────────────────────────────────────
                    4                  
                   x                   
n = 6
    n                                                                    
10⋅x ⋅(a⋅sin(log(x)) + 4⋅a⋅cos(log(x)) + 4⋅b⋅sin(log(x)) - b⋅cos(log(x)))
─────────────────────────────────────────────────────────────────────────
                                     5                                   
                                    x                                    
n = 7
    n                                                                          
10⋅x ⋅(-9⋅a⋅sin(log(x)) - 19⋅a⋅cos(log(x)) - 19⋅b⋅sin(log(x)) + 9⋅b⋅cos(log(x))
───────────────────────────────────────────────────────────────────────────────
                                        6                                      
                                       x                                       

 
)
─
 
 
n = 8
    n                                                                          
10⋅x ⋅(73⋅a⋅sin(log(x)) + 105⋅a⋅cos(log(x)) + 105⋅b⋅sin(log(x)) - 73⋅b⋅cos(log(
───────────────────────────────────────────────────────────────────────────────
                                          7                                    
                                         x                                     

    
x)))
────
    
    
n = 9
    n                                                                          
20⋅x ⋅(-308⋅a⋅sin(log(x)) - 331⋅a⋅cos(log(x)) - 331⋅b⋅sin(log(x)) + 308⋅b⋅cos(l
───────────────────────────────────────────────────────────────────────────────
                                           8                                   
                                          x                                    

       
og(x)))
───────
       
       
n = 10
      n                                                                        
1300⋅x ⋅(43⋅a⋅sin(log(x)) + 36⋅a⋅cos(log(x)) + 36⋅b⋅sin(log(x)) - 43⋅b⋅cos(log(
───────────────────────────────────────────────────────────────────────────────
                                          9                                    
                                         x                                     

    
x)))
────
    
    
$

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数学 - Python - 解析学 - 各種の初等関数 - 三角関数(続き)、逆三角関数(高階微分、帰納法)

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