数学 - Python - 解析学 - 各種の初等関数 - 三角関数(続き)、逆三角関数 - 正弦と余弦、指数関数、微分、等式

解析入門(上)
楽天ブックス Yahoo!

学習環境

• Surface、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro 10.5 + Apple Pencil
• MyScript Nebo - MyScript(iPad アプリ(iOS))
• 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 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 Yahoo!)

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の解答を求めてみる。

4. 
   
   $\begin{array}{l}\frac{d}{\mathrm{dx}}\left(e^x\sin x\right)\\ =e^x\sin x+e^x\cos x\\ =e^x\left(\sin x+\cos x\right)\\ =e^x\sqrt{2}\left(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x\right)\\ ={\left(\sqrt{2}\right)}^1{e}^x\left(\sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4}\right)\\ ={\left(\sqrt{2}\right)}^1{e}^x\sin \left(x+\frac{1·\pi }{4}\right)\end{array}$

   また、

   $\begin{array}{l}\frac{d^n}{{\mathrm{dx}}^n}\left(e^x\sin x\right)\\ =\frac{d}{\mathrm{dx}}\left({\left(\sqrt{2}\right)}^{n-1}{e}^x\sin \left(x+\frac{\left(n-1\right)\pi }{4}\right)\right)\\ ={\left(\sqrt{2}\right)}^{n-1}\left(e^x\sin \left(x+\frac{\left(n-1\right)\pi }{4}\right)+e^x\cos \left(x+\frac{\left(n-1\right)\pi }{4}\right)\right)\\ ={\left(\sqrt{2}\right)}^{n-1}{e}^x\left(\sin \left(x+\frac{\left(n-1\right)\pi }{4}\right)+\cos \left(x+\frac{\left(n-1\right)\pi }{4}\right)\right)\\ ={\left(\sqrt{2}\right)}^n{e}^x\left(\frac{1}{\sqrt{2}}\sin \left(x+\frac{\left(n-1\right)\pi }{4}\right)+\frac{1}{\sqrt{2}}\cos \left(x+\frac{\left(n-1\right)\pi }{4}\right)\right)\\ ={\left(\sqrt{2}\right)}^n{e}^x\left(\sin \left(x+\frac{\left(n-1\right)\pi }{4}\right)\cos \frac{\pi }{4}+\sin \frac{\pi }{4}\cos \left(x+\frac{\left(n-1\right)\pi }{4}\right)\right)\\ ={\left(\sqrt{2}\right)}^n{e}^x\sin \left(x+\frac{\left(n-1\right)\pi }{4}+\frac{\pi }{4}\right)\\ ={\left(\sqrt{2}\right)}^n{e}^x\sin \left(x+\frac{n\pi }{4}\right)\end{array}$

   よって帰納法により 成り立つ。

   （証明終）

コード

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, sin, exp, Derivative, pi, plot

print('4.')

x = symbols('x')
n = symbols('n', integer=True)
f = exp(x) * sin(x)
d = Derivative(f, x, 1)
d1 = d.doit()
g = sqrt(2) ** n * exp(x) * sin(x + n * pi / 4)

for o in [d, d1]:
    pprint(o.factor())
    print()

ns = range(1, 6)
p = plot(*[d1.subs({n: n0}) for n0 in ns],
         *[g.subs({n: n0}) for n0 in ns],
         (x, -5, 5),
         ylim=(-5, 5),
         show=False,
         legend=False)

colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']

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

for t in zip(p, colors):
    pprint(t)
    print()
p.show()
p.save('sample4.png')

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

C:\Users\...>py sample4.py
4.
d ⎛ x       ⎞
──⎝ℯ ⋅sin(x)⎠
dx           

                   x
(sin(x) + cos(x))⋅ℯ 

(cartesian line: exp(x)*sin(x) + exp(x)*cos(x) for x over (-5.0, 5.0), red)

(cartesian line: exp(x)*sin(x) + exp(x)*cos(x) for x over (-5.0, 5.0), green)

(cartesian line: exp(x)*sin(x) + exp(x)*cos(x) for x over (-5.0, 5.0), blue)

(cartesian line: exp(x)*sin(x) + exp(x)*cos(x) for x over (-5.0, 5.0), brown)

(cartesian line: exp(x)*sin(x) + exp(x)*cos(x) for x over (-5.0, 5.0), orange)

(cartesian line: sqrt(2)*exp(x)*sin(x + pi/4) for x over (-5.0, 5.0), purple)

(cartesian line: 2*exp(x)*cos(x) for x over (-5.0, 5.0), pink)

(cartesian line: 2*sqrt(2)*exp(x)*cos(x + pi/4) for x over (-5.0, 5.0), gray)

(cartesian line: -4*exp(x)*sin(x) for x over (-5.0, 5.0), skyblue)

(cartesian line: -4*sqrt(2)*exp(x)*sin(x + pi/4) for x over (-5.0, 5.0), yello
w)


C:\Users\...>