File:Integration path in complex plane crossing saddle-point.png
Summary
| Description |
English: A modular surface (blue) with an integration path (red and green) which crosses a saddle-point. The modular surface describes . The red part of the integration path is the part which crosses the saddle-point. The green part is the "tail". |
| Date | |
| Source | Own work |
| Author | Dom walden |
| PNG development | |
| Source code | Python codeimport 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/4, np.pi/4, 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, 'red')
theta = np.linspace(np.pi - np.pi*3/4, np.pi + np.pi*3/4, 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 the "tail"
ax.plot(x, y, z, 'green')
ax.legend()
plt.show()
|
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