## 2019年12月12日木曜日

### 数学 - Python - 解析学 - 積分法 - 不定積分、広義積分 - 有理関数、共通因数を持たない整式、発散

1. $\begin{array}{l}Q\left(x\right)={\left(x-\gamma \right)}^{k}Q\text{'}\left(x\right)\\ k\ge 1\\ Q\text{'}\left(\gamma \right)\ne 0\end{array}$

とおけば、

$\begin{array}{l}{\left(x-\gamma \right)}^{k}R\left(x\right)\\ ={\left(x-r\right)}^{k}\frac{P\left(x\right)}{Q\left(x\right)}\\ ={\left(x-\gamma \right)}^{k}\frac{P\left(x\right)}{{\left(x-\gamma \right)}^{k}Q\text{'}\left(x\right)}\\ =\frac{P\left(x\right)}{Q\text{'}\left(x\right)}\end{array}$

よって、 整式 P、 Q は共通国数を持たないことから

$\begin{array}{l}\underset{x\to \gamma }{\mathrm{lim}}{\left(x-\gamma \right)}^{k}R\left(x\right)\\ =\frac{P\left(\gamma \right)}{Q\text{'}\left(\gamma \right)}\\ \ne 0\end{array}$

また、

$\underset{x\to \gamma }{\mathrm{lim}}{\left(x-\gamma \right)}^{k}=0$

よって、

${\int }_{a}^{b}R\left(x\right)\mathrm{dx}$

は収束しない。

（証明終）

コード

#!/usr/bin/env python3
from sympy import pprint, symbols, plot, Integral

print('9.')

a = -5
b = 5
x = symbols('x')
p = (x - 2) * (x - 3)
q = x * (x - 1) * (x - 4)
r = p / q
I = Integral(r, (x, a, b))

for o in [I, I.doit()]:
pprint(o)
print()

p = plot((r, (x, -5, -0.00001)),
(r, (x, 0.00001, 0.99999)),
(r, (x, 1.00001, 3.99999)),
(r, (x, 4.000001, 5)),
ylim=(-5, 5),
show=False,
legend=True)
colors = ['red', 'green', 'blue', 'brown', 'orange',
'purple', 'pink', 'gray', 'skyblue', 'yellow']

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

p.show()
p.save('sample9.png')


% ./sample9.py
9.
5
⌠
⎮   (x - 3)⋅(x - 2)
⎮  ───────────────── dx
⎮  x⋅(x - 4)⋅(x - 1)
⌡
-5

nan

%