## 2019年11月12日火曜日

### 数学 - Python - 解析学 - “ε-δ”その他 - 複素数 - 複素数値関数 - 微分、極形式

1. $\begin{array}{l}F\left(t\right)=f\left(t\right)+ig\left(t\right)\\ f\left(t\right),g\left(t\right)\in \text{ℝ},\end{array}$

とおく。
このとき、

$\begin{array}{l}F\text{'}\left(t\right)=f\text{'}\left(t\right)+ig\text{'}\left(t\right)\\ \frac{d}{\mathrm{dt}}{e}^{F\left(t\right)}\\ =\frac{d}{\mathrm{dt}}\left({e}^{f\left(t\right)+ig\left(t\right)}\right)\\ =\frac{d}{\mathrm{dt}}{e}^{f\left(t\right)}{e}^{ig\left(t\right)}\\ ={e}^{f\left(t\right)}f\text{'}\left(t\right){e}^{ig\left(t\right)}+{e}^{f\left(t\right)}\frac{d}{\mathrm{dt}}\left(\mathrm{cos}\left(g\left(t\right)\right)+i\mathrm{sin}\left(g\left(t\right)\right)\right)\\ =f\text{'}\left(t\right){e}^{f\left(t\right)+ig\left(t\right)}+{e}^{f\left(t\right)}\left(-\mathrm{sin}\left(g\left(t\right)\right)g\text{'}\left(t\right)+i\left(cos\left(g\left(t\right)\right)\right)g\text{'}\left(t\right)\right)\\ =f\text{'}\left(t\right){e}^{F\left(t\right)}+{e}^{f\left(t\right)}g\text{'}\left(t\right)\left(-\mathrm{sin}\left(g\left(t\right)\right)+icos\left(g\left(t\right)\right)\right)\\ =f\text{'}\left(t\right){e}^{F\left(t\right)}+g\text{'}\left(t\right){e}^{f\left(t\right)}i\left(\mathrm{cos}\left(g\left(t\right)\right)+i\mathrm{sin}\left(g\left(t\right)\right)\right)\\ =f\text{'}\left(t\right){e}^{F\left(t\right)}+g\text{'}\left(t\right){e}^{f\left(t\right)}{e}^{ig\left(t\right)}\\ =f\text{'}\left(t\right){e}^{F\left(t\right)}+g\text{'}\left(t\right){e}^{f\left(t\right)+ig\left(t\right)}\\ =f\text{'}\left(t\right){e}^{F\left(x\right)}+g\text{'}\left(t\right){e}^{F\left(t\right)}\\ =\left(f\text{'}\left(t\right)+g\text{'}\left(t\right)\right){e}^{F\left(t\right)}\\ =F\text{'}\left(t\right){e}^{F\left(t\right)}\end{array}$

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import pprint, symbols, cos, sin, exp, I, Derivative

print('1.')

t = symbols('t', real=True)
f = cos(t) + I * sin(t)

class MyTestCase(TestCase):
def test(self):
self.assertEqual(Derivative(exp(f)).doit(),
Derivative(f, t, 1).doit() * exp(f))

if __name__ == '__main__':
main()


% ./sample1.py -v
1.
test (__main__.MyTestCase) ... ok

----------------------------------------------------------------------
Ran 1 test in 0.006s

OK
%