import matplotlib.pyplot as plt
from matplotlib import cm
import numpy as np

# Make data
X = np.arange(-4, 4, 0.25)
Y = np.arange(-4, 4, 0.25)
X, Y = np.meshgrid(X, Y)
Z = abs(np.exp(X+Y*1j) / (1 - (X+Y*1j)))

# Plot the surface
fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
ax.plot_wireframe(X, Y, Z, alpha=0.5)

ax.set(xticklabels=[],
       yticklabels=[],
       zticklabels=[])

# Prepare arrays x, y, z
theta = np.linspace(-np.pi, np.pi, 1000)
r = 2
z = abs(np.exp(r * np.exp(1j*theta)) / (1 - r * np.exp(1j*theta)))
x = r * np.sin(theta + np.pi/2)
y = r * np.cos(theta + np.pi/2)

# Plot through saddle-point
ax.plot(x, y, z, 'green')

ax.legend()

plt.show()