python/Matplotlib绘制复变函数图像教程

今天发现sympy依赖的库mpmath里也有很多数学函数，其中也有在复平面绘制二维图的函数cplot，具体例子如下

from mpmath import *

def f1(z):
 return z

def f2(z):
 return z**3

def f3(z):
 return (z**4-1)**(1/4)

def f4(z):
 return 1/z

def f5(z):
 return atan(z)

def f6(z):
 return sqrt(z)

cplot(f1)
cplot(f2)
cplot(f3)
cplot(f4)
cplot(f5)
cplot(f6)

参照matlab绘制复变函数的例子，使用python实现绘制复变函数图像，网上还没搜到相关的文章，在这里分享出来供大家学习。

'''
参照matlab绘制复变函数的例子，创建函数cplxgrid,cplxmap,cplxroot
'''
# 1.导入相关库
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import *

# 2.创建函数
def cplxgrid(m):
 '''Return polar coordinate complex grid.

 Parameters
 ----------
 m: int

 Returns
 ----------
 z: ndarray,with shape (m+1)-by-(2*(m+1))
 '''
 m = m
 r = np.arange(0,m).reshape(m,1) / m
 theta = np.pi * np.arange(-m,m) / m
 z = r * np.exp(1j * theta)

 return z

def cplxroot(n=3,m=20):
 '''
 cplxroot(n): renders the Riemann surface for the n-th root
 cplxroot(): renders the Riemann surface for the cube root.
 cplxroot(n,m): uses an m-by-m grid. Default m = 20.

 Use polar coordinates, (r,theta).
 Use polar coordinates, (r,theta).

 Parameters
 ----------
 n: n-th root
 m: int

 Returns
 ----------
 None: Plot the Riemann surface
 '''
 m = m+1
 r = np.arange(0,m).reshape(m,1) / m
 theta = np.pi * np.arange(-n * m, n * m) / m
 z = r * np.exp(1j * theta)
 s = r * (1/n) * np.exp(1j * theta / n)
 fig = plt.figure()
 ax = fig.add_subplot(111,projection='3d')
 # ax.plot_surface(np.real(z),np.imag(z),np.real(s),color = np.imag(s))
 ax.plot_surface(np.real(z),np.imag(z),np.real(s),cmap = plt.cm.hsv)
 ax.set_xlim((-1,1))
 ax.set_ylim((-1,1))
 ax.set_xlabel('Real')
 ax.set_ylabel('Imag')
 ax.set_xticks([])
 ax.set_yticks([])
 ax.set_zticks([])
 ax.set_autoscalez_on(True)#z轴自动缩放 
 ax.grid('on')
 plt.show()

def cplxmap(z,cfun):
 '''
 Plot a function of a complex variable.

 Parameters
 ----------
 z: complex plane
 cfun: complex function to plot

 Returns
 ----------
 None: Plot the surface of complex function
 '''
 blue = 0.2
 x = np.real(z)
 y = np.imag(z)
 u = np.real(cfun)
 v = np.imag(cfun)
 M = np.max(np.max(u))#复变函数实部最大值
 m = np.min(np.min(u))#复变函数实部最大值
 s = np.ones(z.shape)
 fig = plt.figure()
 ax = fig.add_subplot(111,projection='3d')
 # 投影部分用线框图
 surf1 = ax.plot_wireframe(x,y,m*s,cmap=plt.cm.hsv)
 surf2 = ax.plot_surface(x,y,u,cmap=plt.cm.hsv)

 #绘制复变函数1/z时会出错，ValueError: Axis limits cannot be NaN or Inf
 # ax.set_zlim(m, M) 
 ax.set_xlim((-1,1))
 ax.set_ylim((-1,1))
 ax.set_xlabel('Real')
 ax.set_ylabel('Imag')
 ax.set_xticks([])
 ax.set_yticks([])
 ax.set_zticks([])
 ax.set_autoscalez_on(True)#z轴自动缩放

 ax.grid('on')
 plt.show()

def _test_cplxmap():
 '''测试cplxmap函数'''
 z = cplxgrid(30)
 w1 = z
 w2 = z**3
 w3 = (z**4-1)**(1/4)
 w4 = 1/z
 w5 = np.arctan(2*z)
 w6 = np.sqrt(z)
 w = [w1,w2,w3,w4,w5,w6]
 for i in w:
 cplxmap(z,i)

def _test_cplxroot():
 '''测试cplxroot函数'''
 cplxroot(n=2)
 cplxroot(n=3)
 cplxroot(n=4)
 cplxroot(n=5)

if __name__ == '__main__':
 _test_cplxmap()
 _test_cplxroot()