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.

Complex Numbers Review¶

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

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¶

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¶

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¶

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¶

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

chf = np.exp(1j * mu * t - sigma**2 * 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=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.

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 - sigma**2 * t**2 / 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.

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 - sigma**2 * t**2 / 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¶

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 - sigma**2 * t**2 / 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¶

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 - sigma**2 * t**2 / 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¶

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

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¶

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

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

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

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¶

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¶

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¶

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¶

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¶

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 - sigma**2 * t**2 / 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