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

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。

$\begin{array}{l} \frac{d}{dx} (e^{x} \sin x) \\ = e^{x} \sin x + e^{x} \cos x \\ = e^{x} (\sin x + \cos x) \\ = e^{x} \sqrt{2} (\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x) \\ = {(\sqrt{2})}^{1} e^{x} (\sin x \cos \frac{π}{4} + \cos x \sin \frac{π}{4}) \\ = {(\sqrt{2})}^{1} e^{x} \sin (x + \frac{1 \cdot π}{4}) \end{array}$
また、
$\begin{array}{l} \frac{d^{n}}{{dx}^{n}} (e^{x} \sin x) \\ = \frac{d}{dx} ({(\sqrt{2})}^{n - 1} e^{x} \sin (x + \frac{(n - 1) π}{4})) \\ = {(\sqrt{2})}^{n - 1} (e^{x} \sin (x + \frac{(n - 1) π}{4}) + e^{x} \cos (x + \frac{(n - 1) π}{4})) \\ = {(\sqrt{2})}^{n - 1} e^{x} (\sin (x + \frac{(n - 1) π}{4}) + \cos (x + \frac{(n - 1) π}{4})) \\ = {(\sqrt{2})}^{n} e^{x} (\frac{1}{\sqrt{2}} \sin (x + \frac{(n - 1) π}{4}) + \frac{1}{\sqrt{2}} \cos (x + \frac{(n - 1) π}{4})) \\ = {(\sqrt{2})}^{n} e^{x} (\sin (x + \frac{(n - 1) π}{4}) \cos \frac{π}{4} + \sin \frac{π}{4} \cos (x + \frac{(n - 1) π}{4})) \\ = {(\sqrt{2})}^{n} e^{x} \sin (x + \frac{(n - 1) π}{4} + \frac{π}{4}) \\ = {(\sqrt{2})}^{n} e^{x} \sin (x + \frac{n π}{4}) \end{array}$
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\...>

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2019年9月15日日曜日

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

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