## 2019年4月4日木曜日

### 数学 - Python - 解析学 - 数 - 複素数(絶対値、不等式の証明、共役、除算、分数)

1. $\begin{array}{}{\left|\frac{\alpha -\beta }{1-\stackrel{-}{\alpha }\beta }\right|}^{2}\\ =\frac{{\left|\alpha -\beta \right|}^{2}}{{\left|1-\stackrel{-}{\alpha }\beta \right|}^{2}}\\ =\frac{\left(\alpha -\beta \right)\left(\stackrel{-}{\alpha -\beta }\right)}{\left(1-\stackrel{-}{\alpha }\beta \right)\stackrel{-}{\left(1-\stackrel{-}{\alpha }\beta \right)}}\\ =\frac{\left(\alpha -\beta \right)\left(\stackrel{-}{\alpha }-\stackrel{-}{\beta }\right)}{\left(1-\stackrel{-}{\alpha }\beta \right)\left(1-\alpha \stackrel{-}{\beta }\right)}\\ =\frac{{\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}-\alpha \stackrel{-}{\beta }-\stackrel{-}{\alpha }\beta }{1+{\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}-\alpha \stackrel{-}{\beta }-\stackrel{-}{\alpha }\beta }\\ \left(1+{\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}-\alpha \stackrel{-}{\beta }-\stackrel{-}{\alpha }\beta \right)-\left({\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}-\alpha \stackrel{-}{\beta }-\stackrel{-}{\alpha }\beta \right)\\ =1+{\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}-{\left|\alpha \right|}^{2}-{\left|\beta \right|}^{2}\\ =\left(1-{\left|\alpha \right|}^{2}\right)\left(1-{\left|\beta \right|}^{2}\right)\\ >0\\ {\left|\frac{\alpha -\beta }{1-\stackrel{-}{\alpha }\beta }\right|}^{2}<1\end{array}$

よって、

$\left|\frac{\alpha -\beta }{1-\stackrel{-}{\alpha }\beta }\right|<1$

（証明終）

コード

Python 3

#!/usr/bin/env python3
from sympy import symbols,  pprint, I, Rational, sqrt
import random

print('8.')

alpha = Rational(1, 3) + sqrt(3) / 2 * I
beta = 1 / sqrt(2) + 1 / (2 * sqrt(2)) * I
z = (alpha - beta) / (1 - alpha.conjugate() * beta)
for o in [z, abs(z), float(abs(z))]:
pprint(o)
print()


C:\Users\...>py sample8.py
8.
√2   1   √2⋅ⅈ   √3⋅ⅈ
- ── + ─ - ──── + ────
2    3    4      2
──────────────────────────
⎛1   √3⋅ⅈ⎞ ⎛√2   √2⋅ⅈ⎞
1 - ⎜─ - ────⎟⋅⎜── + ────⎟
⎝3    2  ⎠ ⎝2     4  ⎠

_________________
╱   √6   √2   107
╱  - ── - ── + ───
╲╱     4    3     72
─────────────────────
_________________
╱   √6   √2   443
╱  - ── - ── + ───
╲╱     4    3    288

0.940948669874474

C:\Users\...>