泰勒展开&傅里叶展开
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\begin{aligned} f(z)& = \frac{1}{1-z}\\ &=1+z+z^2+z^3+...\\ &=1 + re^{\theta} + r^{2}e^{2\theta}+ r^{3}e^{3\theta}+... \end{aligned}
f(z)=1−z1=1+z+z2+z3+...=1+reθ+r2e2θ+r3e3θ+...
复反演
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f(z) = \frac{1}{z}\\
f(z)=z1
模长变成倒数，角度变成相反数
一个过原点的直线经过复反演之后还是一条直线
一个直线不经过原点，反演之后变成一个圆
一个经过原点的圆反演之后是一个直线
一个不经过原点的圆反演之后是一个圆
两个例子：
复变函数的积分
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\int_{ z_{0} }^{ z_{1} }f(z)dz\\ =\sum f(\Delta z) \Delta z
∫z0z1f(z)dz=∑f(Δz)Δz
每一小段旋转拉伸之后的加和
向量加法，最终的位置减去起始位置就是积分函数值
f ( z ) = 1 z f ( a + b i ) = 1 a + b i = a − b i a 2 + b 2 \begin{aligned} f(z) &= \frac{1}{z} \\f(a+bi) &= \frac{1}{a+bi} \\ \\&= \frac{a-bi}{a^2+b^2} \end{aligned} f(z)f(a+bi)=z1=a+bi1=a2+b2a−bi
1个例子
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
class shape:
x = []
y = []
def __init__(self, x, y):
self.x = x.copy()
self.y = y.copy()
def draw(shapetemp):
plt.plot(shapetemp.x,shapetemp.y)
def fanyan(shape1):
temp = []
for x,y in zip(shape1.x,shape1.y):
temp.append( (x/(x*x+y*y),-y/(x*x+y*y)) )
# print((x/(x*x+y*y),-y/(x*x+y*y)) )
return shape(list(list(zip(*temp))[0]),list(list(zip(*temp))[1]))
x = np.arange(-10,10,0.001)
y = x+ 0.1
y1 = -x+ 0.1
shape1 = shape(x,y)
shape1 = shape(x,y1)
draw(shape1)
draw(shape2)
sf1 = fanyan(shape1)
sf2 = fanyan(shape2)
draw(sf1)
draw(sf2)
plt.grid()
plt.axis('equal')
plt.show()
