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Complex Function Visualization

This document covers techniques for visualizing complex-valued functions in 3D, with emphasis on characteristic functions from probability theory and Euler's formula.

Mental Model

A complex function maps a real input to a complex output with two parts (real and imaginary), so you need three dimensions to see it: one for the input and two for the output. Plot the real part as one surface and the imaginary part as another, or use a single surface colored by phase. This makes abstract complex analysis tangible.

Mathematical Visualization Pipeline

Every 3D visualization in this section follows the same pipeline:

  1. Define the mathematical object --- function, distribution, parametric curve
  2. Choose a representation --- curve (1D to 3D), surface (2D to 3D), or color-coded
  3. Map variables to axes --- input to x, real part to y, imaginary part to z
  4. Choose visual encoding --- height = magnitude, color = phase, separate real/imag
  5. Choose viewpoint --- view_init(elev, azim) to reveal specific structure

The goal: reveal structure that is invisible in equations alone.

Complex Numbers Review

A complex number \(z = x + iy\) has real part \(x\) and imaginary part \(y\). In Python:

```python import numpy as np

Creating complex numbers

z = complex(3, 4) # 3 + 4i z = 3 + 4j # Python notation z = np.exp(1j * np.pi) # Euler's formula: e^{iπ} = -1

Accessing parts

z.real # Real part z.imag # Imaginary part np.abs(z) # Magnitude |z| np.angle(z) # Phase angle ```

Euler's Formula

\[e^{i\theta} = \cos(\theta) + i\sin(\theta)\]

1. Basic Visualization

```python import matplotlib.pyplot as plt import numpy as np

t = np.linspace(0, 2 * np.pi, 500) z = np.exp(1j * t)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(t, np.real(z), np.imag(z), 'blue', linewidth=2) ax.set_xlabel('θ') ax.set_ylabel('cos(θ)') ax.set_zlabel('sin(θ)') ax.set_title("Euler's Formula: \(e^{i\\theta}\)") ax.view_init(25, -60) plt.show() ```

2. Multiple Rotations

```python t = np.linspace(0, 6 * np.pi, 1000) z = np.exp(1j * t)

fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'}) ax.plot(t, np.real(z), np.imag(z), 'blue', linewidth=1.5) ax.set_xlabel('θ') ax.set_ylabel('Re\((e^{i\\theta})\)') ax.set_zlabel('Im\((e^{i\\theta})\)') ax.set_title("Euler's Formula: Helix in Complex Space") ax.view_init(20, -70) plt.show() ```

3. Frequency Comparison

```python t = np.linspace(0, 4 * np.pi, 500)

fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

frequencies = [1, 2, 3] colors = ['blue', 'green', 'red']

for ax, freq, color in zip(axes, frequencies, colors): z = np.exp(1j * freq * t) ax.plot(t, np.real(z), np.imag(z), color, linewidth=1.5) ax.set_title(f'\(e^{{i \\cdot {freq} \\cdot t}}\)') ax.set_xlabel('t') ax.set_ylabel('Re') ax.set_zlabel('Im') ax.view_init(25, -60)

plt.tight_layout() plt.show() ```

Characteristic Functions

The characteristic function of a random variable \(X\) is defined as:

\[\varphi_X(t) = E[e^{itX}]\]

Normal Distribution

For \(X \sim N(\mu, \sigma^2)\):

\[\varphi_{N(\mu,\sigma^2)}(t) = e^{i\mu t - \frac{1}{2}\sigma^2 t^2}\]

1. Standard Normal Characteristic Function

```python from scipy import stats

t = np.linspace(-10, 10, 1000) chf = np.exp(-t**2 / 2) # Standard normal: μ=0, σ=1

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(t, np.real(chf), np.imag(chf), 'blue', linewidth=2) ax.set_xlabel('t') ax.set_ylabel('Re(φ)') ax.set_zlabel('Im(φ)') ax.set_title('Standard Normal: \(\\varphi(t) = e^{-t^2/2}\)') ax.view_init(30, -120) plt.show() ```

2. General Normal Characteristic Function

```python mu = 3.0 sigma = 2.6 t = np.linspace(-15, 15, 1000)

chf = np.exp(1j * mu * t - sigma2 * t2 / 2)

fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'}) ax.plot(t, np.real(chf), np.imag(chf), 'blue', linewidth=1.5) ax.set_xlabel('t') ax.set_ylabel('Re(φ)') ax.set_zlabel('Im(φ)') ax.set_title(f'Normal CF: \(\\mu={mu}\), \(\\sigma={sigma}\)') ax.view_init(30, -120) plt.show() ```

3. Effect of Mean (μ)

Changing μ affects the oscillation frequency.

```python t = np.linspace(-10, 10, 1000) sigma = 1

fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

mus = [0, 2, 4]

for ax, mu in zip(axes, mus): chf = np.exp(1j * mu * t - sigma2 * t2 / 2) ax.plot(t, np.real(chf), np.imag(chf), 'blue', linewidth=1.5) ax.set_title(f'μ = {mu}, σ = {sigma}') ax.set_xlabel('t') ax.set_ylabel('Re') ax.set_zlabel('Im') ax.view_init(30, -120)

plt.tight_layout() plt.show() ```

4. Effect of Standard Deviation (σ)

Changing σ affects the decay rate.

```python t = np.linspace(-10, 10, 1000) mu = 0

fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

sigmas = [0.5, 1, 2]

for ax, sigma in zip(axes, sigmas): chf = np.exp(1j * mu * t - sigma2 * t2 / 2) ax.plot(t, np.real(chf), np.imag(chf), 'green', linewidth=1.5) ax.set_title(f'μ = {mu}, σ = {sigma}') ax.set_xlabel('t') ax.set_ylabel('Re') ax.set_zlabel('Im') ax.view_init(30, -120)

plt.tight_layout() plt.show() ```

Complete Distribution Visualization

Visualize PDF, CDF, and characteristic function together.

1. Standard Normal

```python mu = 0 sigma = 1 x = np.linspace(-4, 4, 200) t = np.linspace(-10, 10, 1000)

pdf = stats.norm(mu, sigma).pdf(x) cdf = stats.norm(mu, sigma).cdf(x) chf = np.exp(1j * mu * t - sigma2 * t2 / 2)

fig = plt.figure(figsize=(15, 5))

PDF

ax1 = fig.add_subplot(1, 3, 1) ax1.plot(x, pdf, 'steelblue', linewidth=2) ax1.fill_between(x, pdf, alpha=0.3) ax1.set_xlabel('x') ax1.set_ylabel('f(x)') ax1.set_title('PDF') ax1.grid(alpha=0.3)

CDF

ax2 = fig.add_subplot(1, 3, 2) ax2.plot(x, cdf, 'darkgreen', linewidth=2) ax2.set_xlabel('x') ax2.set_ylabel('F(x)') ax2.set_title('CDF') ax2.grid(alpha=0.3)

Characteristic Function

ax3 = fig.add_subplot(1, 3, 3, projection='3d') ax3.plot(t, np.real(chf), np.imag(chf), 'purple', linewidth=1.5) ax3.set_xlabel('t') ax3.set_ylabel('Re(φ)') ax3.set_zlabel('Im(φ)') ax3.set_title('Characteristic Function') ax3.view_init(30, -120)

plt.suptitle(f'Standard Normal Distribution (μ={mu}, σ={sigma})', fontsize=14, fontweight='bold') plt.tight_layout() plt.show() ```

2. General Normal

```python mu = 3.0 sigma = 2.6 x = np.linspace(-15, 20, 200) t = np.linspace(-10, 10, 1000)

pdf = stats.norm(mu, sigma).pdf(x) cdf = stats.norm(mu, sigma).cdf(x) chf = np.exp(1j * mu * t - sigma2 * t2 / 2)

fig = plt.figure(figsize=(14, 5))

ax0 = fig.add_subplot(1, 3, 1) ax0.plot(x, pdf, 'steelblue', linewidth=2) ax0.fill_between(x, pdf, alpha=0.3, color='steelblue') ax0.set_xlabel('\(x\)', fontsize=12) ax0.set_ylabel('\(f(x)\)', fontsize=12) ax0.set_title('PDF', fontsize=13) ax0.grid(alpha=0.3)

ax1 = fig.add_subplot(1, 3, 2) ax1.plot(x, cdf, 'darkgreen', linewidth=2) ax1.set_xlabel('\(x\)', fontsize=12) ax1.set_ylabel('\(F(x)\)', fontsize=12) ax1.set_title('CDF', fontsize=13) ax1.grid(alpha=0.3)

ax2 = fig.add_subplot(1, 3, 3, projection='3d') ax2.plot(t, np.real(chf), np.imag(chf), 'darkviolet', linewidth=1.5) ax2.set_xlabel('\(t\)', fontsize=11) ax2.set_ylabel('Re\((\\varphi)\)', fontsize=11) ax2.set_zlabel('Im\((\\varphi)\)', fontsize=11) ax2.set_title('\(\\varphi(t) = e^{i\\mu t - \\sigma^2 t^2/2}\)', fontsize=12) ax2.view_init(25, -130)

plt.suptitle(f'Normal Distribution: \(\\mu={mu}\), \(\\sigma={sigma}\)', fontsize=14, fontweight='bold') plt.tight_layout() plt.show() ```

Other Distribution Characteristic Functions

1. Exponential Distribution

\[\varphi(t) = \frac{\lambda}{\lambda - it}\]

```python lam = 1.0 t = np.linspace(-5, 5, 1000) chf = lam / (lam - 1j * t)

fig = plt.figure(figsize=(14, 5))

PDF

x = np.linspace(0, 6, 200) ax1 = fig.add_subplot(1, 3, 1) ax1.plot(x, stats.expon(scale=1/lam).pdf(x), 'steelblue', linewidth=2) ax1.set_xlabel('x') ax1.set_ylabel('f(x)') ax1.set_title('Exponential PDF') ax1.grid(alpha=0.3)

Real and Imaginary parts

ax2 = fig.add_subplot(1, 3, 2) ax2.plot(t, np.real(chf), 'b-', label='Real', linewidth=2) ax2.plot(t, np.imag(chf), 'r-', label='Imaginary', linewidth=2) ax2.set_xlabel('t') ax2.legend() ax2.set_title('CF Components') ax2.grid(alpha=0.3)

3D

ax3 = fig.add_subplot(1, 3, 3, projection='3d') ax3.plot(t, np.real(chf), np.imag(chf), 'green', linewidth=1.5) ax3.set_xlabel('t') ax3.set_ylabel('Re') ax3.set_zlabel('Im') ax3.set_title('\(\\varphi(t) = \\frac{\\lambda}{\\lambda - it}\)') ax3.view_init(30, -120)

plt.suptitle('Exponential Distribution (λ=1)', fontsize=14, fontweight='bold') plt.tight_layout() plt.show() ```

2. Cauchy Distribution

\[\varphi(t) = e^{-|t|}\]

```python t = np.linspace(-5, 5, 1000) chf = np.exp(-np.abs(t))

fig = plt.figure(figsize=(14, 5))

PDF

x = np.linspace(-10, 10, 500) ax1 = fig.add_subplot(1, 3, 1) ax1.plot(x, stats.cauchy.pdf(x), 'steelblue', linewidth=2) ax1.set_xlabel('x') ax1.set_ylabel('f(x)') ax1.set_title('Cauchy PDF') ax1.set_ylim(0, 0.35) ax1.grid(alpha=0.3)

Components

ax2 = fig.add_subplot(1, 3, 2) ax2.plot(t, np.real(chf), 'b-', label='Real', linewidth=2) ax2.plot(t, np.imag(chf), 'r-', label='Imaginary', linewidth=2) ax2.set_xlabel('t') ax2.legend() ax2.set_title('CF Components') ax2.grid(alpha=0.3)

3D

ax3 = fig.add_subplot(1, 3, 3, projection='3d') ax3.plot(t, np.real(chf), np.imag(chf), 'red', linewidth=2) ax3.set_xlabel('t') ax3.set_ylabel('Re') ax3.set_zlabel('Im') ax3.set_title('\(\\varphi(t) = e^{-|t|}\)') ax3.view_init(30, -120)

plt.suptitle('Cauchy Distribution', fontsize=14, fontweight='bold') plt.tight_layout() plt.show() ```

3. Uniform Distribution

For \(X \sim U(a, b)\):

\[\varphi(t) = \frac{e^{itb} - e^{ita}}{it(b-a)}\]

```python a, b = -1, 1 t = np.linspace(-10, 10, 1000) t_safe = np.where(np.abs(t) < 1e-10, 1e-10, t) # Avoid division by zero chf = (np.exp(1j * t_safe * b) - np.exp(1j * t_safe * a)) / (1j * t_safe * (b - a))

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(t, np.real(chf), np.imag(chf), 'orange', linewidth=1.5) ax.set_xlabel('t') ax.set_ylabel('Re(φ)') ax.set_zlabel('Im(φ)') ax.set_title(f'Uniform CF: U({a}, {b})') ax.view_init(30, -120) plt.show() ```

Damped and Growing Oscillations

1. Damped Oscillation

```python t = np.linspace(0, 10, 500) decay_rate = 0.3 z = np.exp((-decay_rate + 1j) * t)

fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'}) ax.plot(t, np.real(z), np.imag(z), 'green', linewidth=2) ax.set_xlabel('t') ax.set_ylabel('Re') ax.set_zlabel('Im') ax.set_title(f'Damped Oscillation: \(e^{{(-{decay_rate}+i)t}}\)') ax.view_init(25, -60) plt.show() ```

2. Damping Comparison

```python t = np.linspace(0, 8, 500)

fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

decay_rates = [0, 0.2, 0.5] titles = ['No Damping', 'Light Damping', 'Heavy Damping']

for ax, decay, title in zip(axes, decay_rates, titles): z = np.exp((-decay + 1j) * t) ax.plot(t, np.real(z), np.imag(z), linewidth=2) ax.set_title(f'{title}\n\(e^{{(-{decay}+i)t}}\)') ax.set_xlabel('t') ax.set_ylabel('Re') ax.set_zlabel('Im') ax.view_init(25, -60)

plt.tight_layout() plt.show() ```

3D Axes Customization

Pane Colors

```python t = np.linspace(-10, 10, 1000) chf = np.exp(-t**2 / 2)

fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'}) ax.plot(t, np.real(chf), np.imag(chf), 'blue', linewidth=2)

White panes

ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))

ax.set_xlabel('t') ax.set_ylabel('Re(φ)') ax.set_zlabel('Im(φ)') ax.view_init(30, -120) plt.show() ```

Z-Label Rotation

```python fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'}) ax.plot(t, np.real(chf), np.imag(chf), 'blue', linewidth=2)

ax.set_xlabel('t') ax.set_ylabel('Re(φ)') ax.set_zlabel('Im(φ)', rotation=90) ax.zaxis.set_rotate_label(False) ax.view_init(30, -120) plt.show() ```

Publication-Quality Figure

```python mu = 3.0 sigma = 2.6 x = np.linspace(-15, 20, 200) t = np.linspace(-10, 10, 1000)

pdf = stats.norm(mu, sigma).pdf(x) cdf = stats.norm(mu, sigma).cdf(x) chf = np.exp(1j * mu * t - sigma2 * t2 / 2)

fig = plt.figure(figsize=(14, 4.5))

PDF

ax0 = fig.add_subplot(1, 3, 1) ax0.plot(x, pdf, 'steelblue', linewidth=2) ax0.fill_between(x, pdf, alpha=0.25, color='steelblue') ax0.set_xlabel('\(x\)', fontsize=12) ax0.set_ylabel('\(f(x)\)', fontsize=12) ax0.set_title('Probability Density Function', fontsize=12) ax0.grid(alpha=0.3) ax0.tick_params(labelsize=10)

CDF

ax1 = fig.add_subplot(1, 3, 2) ax1.plot(x, cdf, 'darkgreen', linewidth=2) ax1.set_xlabel('\(x\)', fontsize=12) ax1.set_ylabel('\(F(x)\)', fontsize=12) ax1.set_title('Cumulative Distribution Function', fontsize=12) ax1.grid(alpha=0.3) ax1.tick_params(labelsize=10)

Characteristic Function

ax2 = fig.add_subplot(1, 3, 3, projection='3d') ax2.plot(t, np.real(chf), np.imag(chf), 'darkviolet', linewidth=1.5) ax2.set_xlabel('\(t\)', fontsize=11, labelpad=8) ax2.set_ylabel('Re\((\\varphi)\)', fontsize=11, labelpad=8) ax2.set_zlabel('Im\((\\varphi)\)', fontsize=11, labelpad=8) ax2.set_title('Characteristic Function', fontsize=12) ax2.view_init(25, -130)

ax2.w_xaxis.set_pane_color((0.98, 0.98, 0.98, 1.0)) ax2.w_yaxis.set_pane_color((0.98, 0.98, 0.98, 1.0)) ax2.w_zaxis.set_pane_color((0.98, 0.98, 0.98, 1.0)) ax2.tick_params(labelsize=9)

plt.suptitle(f'Normal Distribution: \(\\mu={mu}\), \(\\sigma={sigma}\)', fontsize=14, fontweight='bold', y=1.02) plt.tight_layout() plt.show() ```

Summary Table

Distribution Characteristic Function Notes
Normal \(N(\mu, \sigma^2)\) \(e^{i\mu t - \frac{1}{2}\sigma^2 t^2}\) Oscillates with decay
Standard Normal \(e^{-t^2/2}\) Real-valued, no oscillation
Exponential(\(\lambda\)) \(\frac{\lambda}{\lambda - it}\) Complex rational
Cauchy \(e^{-\|t\|}\) Real-valued
Uniform\((a,b)\) \(\frac{e^{itb} - e^{ita}}{it(b-a)}\) Sinc-like

Exercises

Exercise 1. Visualize the complex function \(f(z) = z^3\) as a 3D surface plot. Create a grid of complex numbers over the square \([-2, 2] \times [-2, 2]\), compute the magnitude \(|f(z)|\) for the height and the phase \(\arg(f(z))\) for the facecolor using the hsv colormap. Add a colorbar labeled "Phase (radians)" and set a descriptive title.

Solution to Exercise 1 
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import Normalize

x = np.linspace(-2, 2, 200)
y = np.linspace(-2, 2, 200)
X, Y = np.meshgrid(x, y)
Z = X + 1j * Y

W = Z ** 3
magnitude = np.abs(W)
phase = np.angle(W)

fig = plt.figure(figsize=(10, 7))
ax = fig.add_subplot(111, projection='3d')

norm = Normalize(vmin=-np.pi, vmax=np.pi)
colors = plt.cm.hsv(norm(phase))

ax.plot_surface(X, Y, magnitude, facecolors=colors, shade=False)
ax.set_xlabel('Re(z)')
ax.set_ylabel('Im(z)')
ax.set_zlabel('|f(z)|')
ax.set_title(r'$f(z) = z^3$: Magnitude Surface with Phase Coloring')

mappable = plt.cm.ScalarMappable(cmap='hsv', norm=norm)
plt.colorbar(mappable, ax=ax, shrink=0.6, label='Phase (radians)')
plt.show()

Exercise 2. Create a domain coloring plot for the function \(f(z) = \frac{1}{z}\) over the region \([-2, 2] \times [-2, 2]\) using imshow. Map the phase angle to the twilight colormap and use the magnitude to modulate the brightness. Clip the magnitude to a maximum of 5 to avoid extreme values near the singularity at \(z = 0\).

Solution to Exercise 2 
import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-2, 2, 500)
y = np.linspace(-2, 2, 500)
X, Y = np.meshgrid(x, y)
Z = X + 1j * Y

W = 1 / Z
phase = np.angle(W)
magnitude = np.abs(W)
magnitude_clipped = np.clip(magnitude, 0, 5)

brightness = magnitude_clipped / 5.0
colors = plt.cm.twilight((phase + np.pi) / (2 * np.pi))
colors[..., :3] *= brightness[..., np.newaxis]

fig, ax = plt.subplots(figsize=(8, 8))
ax.imshow(colors, extent=[-2, 2, -2, 2], origin='lower')
ax.set_xlabel('Re(z)')
ax.set_ylabel('Im(z)')
ax.set_title(r'Domain Coloring of $f(z) = 1/z$')
plt.show()

Exercise 3. Plot the real and imaginary parts of \(f(z) = e^z\) as two separate 3D surface plots side by side using plt.subplots(1, 2, subplot_kw={'projection': '3d'}). Over the domain \([-2, 2] \times [-2, 2]\), show the real part \(e^x \cos(y)\) on the left and the imaginary part \(e^x \sin(y)\) on the right. Use different colormaps for each surface and include titles.

Solution to Exercise 3 
import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-2, 2, 200)
y = np.linspace(-2, 2, 200)
X, Y = np.meshgrid(x, y)
Z = X + 1j * Y

W = np.exp(Z)
real_part = W.real
imag_part = W.imag

fig, axes = plt.subplots(1, 2, figsize=(14, 6),
                          subplot_kw={'projection': '3d'})

axes[0].plot_surface(X, Y, real_part, cmap='viridis', alpha=0.9)
axes[0].set_xlabel('Re(z)')
axes[0].set_ylabel('Im(z)')
axes[0].set_zlabel('Re(f)')
axes[0].set_title(r'Real Part of $e^z$: $e^x \cos(y)$')

axes[1].plot_surface(X, Y, imag_part, cmap='plasma', alpha=0.9)
axes[1].set_xlabel('Re(z)')
axes[1].set_ylabel('Im(z)')
axes[1].set_zlabel('Im(f)')
axes[1].set_title(r'Imaginary Part of $e^z$: $e^x \sin(y)$')

plt.tight_layout()
plt.show()