analytic_continuation.ipynb

Kernel: Python 3

Analytic Continuation

Theoretical Foundation

Analytic continuation is a fundamental technique in complex analysis that extends the domain of a given analytic function beyond its original region of definition. This powerful concept allows us to define functions on larger domains while preserving their analytic properties.

Definition

Let $f(z)$ be an analytic function defined on a domain $D_1 \subset \mathbb{C}$. An analytic continuation of $f$ to a larger domain $D_2 \supset D_1$ is an analytic function $F(z)$ defined on $D_2$ such that:

$$F(z) = f(z) \quad \text{for all } z \in D_1$$

Identity Theorem

The Identity Theorem guarantees uniqueness: If two analytic functions agree on a domain $D_1$, and both can be analytically continued to $D_2$, then their continuations must be identical on $D_2$. This is a consequence of the fact that an analytic function is uniquely determined by its values on any set with an accumulation point.

Power Series Representation

A common method for analytic continuation uses power series. If $f(z)$ has a power series expansion around $z_0$:

$$f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$$

with radius of convergence $R$, then $f$ is analytic within the disk $|z - z_0| < R$. To extend beyond this disk, we choose a new center $z_1$ within the original disk and compute a new power series:

$$f(z) = \sum_{n=0}^{\infty} b_n (z - z_1)^n$$

This process can be repeated to extend the function along any path in the complex plane, provided we avoid singularities.

Natural Boundary

Not all functions can be continued everywhere. A natural boundary is a curve beyond which analytic continuation is impossible. For example, the function:

$$f(z) = \sum_{n=0}^{\infty} z^{2^n}$$

has the unit circle $|z| = 1$ as a natural boundary.

The Riemann Zeta Function Example

The most famous example is the Riemann zeta function:

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$

This series converges only for $\text{Re}(s) > 1$, but through analytic continuation, $\zeta(s)$ can be extended to the entire complex plane (except for a simple pole at $s = 1$). The functional equation:

$$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$$

relates values in the half-plane $\text{Re}(s) < 0$ to those in $\text{Re}(s) > 1$.

Monodromy and Multi-valued Functions

Analytic continuation around singularities can lead to multi-valued functions. Consider $f(z) = \sqrt{z}$. Starting at $z = 1$ with $f(1) = 1$ and analytically continuing around the origin, we return to $z = 1$ but with $f(1) = -1$. This phenomenon is called monodromy and is handled using Riemann surfaces.

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import gamma, zeta
import matplotlib.patches as mpatches
from matplotlib.patches import Circle, FancyArrowPatch
from matplotlib import cm

Computational Example 1: Power Series Continuation

We demonstrate analytic continuation using overlapping power series. Consider the function:

$$f(z) = \frac{1}{1-z}$$

This has a power series around $z_0 = 0$:

$$f(z) = \sum_{n=0}^{\infty} z^n \quad \text{for } |z| < 1$$

To extend beyond the unit circle, we create a new expansion around $z_1 = \frac{1}{2}$.

def power_series_at_origin(z, n_terms=20):
    """Compute 1/(1-z) using power series around z=0"""
    result = np.zeros_like(z, dtype=complex)
    for n in range(n_terms):
        result += z**n
    return result

def power_series_at_half(z, n_terms=20):
    """Compute 1/(1-z) using power series around z=1/2"""
    # 1/(1-z) = 2 * sum_{n=0}^{infty} (2(z-1/2))^n for |z-1/2| < 1/2
    w = z - 0.5
    result = np.zeros_like(z, dtype=complex)
    for n in range(n_terms):
        result += 2 * (2*w)**n
    return result

def exact_function(z):
    """Exact value of 1/(1-z)"""
    return 1.0 / (1.0 - z)

Computational Example 2: Riemann Zeta Function

We'll visualize the Riemann zeta function in the complex plane, demonstrating its analytic continuation from $\text{Re}(s) > 1$ to the entire complex plane.

def plot_zeta_magnitude():
    """Plot magnitude of Riemann zeta function in complex plane"""
    # Create grid
    x = np.linspace(-2, 4, 400)
    y = np.linspace(-15, 15, 400)
    X, Y = np.meshgrid(x, y)
    Z = X + 1j*Y
    
    # Compute zeta function (scipy handles analytic continuation)
    zeta_values = np.zeros_like(Z, dtype=complex)
    
    for i in range(Z.shape[0]):
        for j in range(Z.shape[1]):
            try:
                # Avoid the pole at s=1
                if np.abs(Z[i,j] - 1.0) > 0.1:
                    zeta_values[i,j] = zeta(Z[i,j])
                else:
                    zeta_values[i,j] = np.nan
            except:
                zeta_values[i,j] = np.nan
    
    magnitude = np.abs(zeta_values)
    # Cap very large values for visualization
    magnitude = np.minimum(magnitude, 10)
    
    return X, Y, magnitude, zeta_values

Computational Example 3: Logarithm and Branch Cuts

The complex logarithm $\log(z)$ is multi-valued. We demonstrate how analytic continuation around the origin leads to different branches:

$$\log(z) = \ln|z| + i\arg(z)$$

where $\arg(z)$ is determined modulo $2\pi$.

def demonstrate_logarithm_branches():
    """Visualize different branches of the complex logarithm"""
    theta = np.linspace(0, 4*np.pi, 1000)  # Two full rotations
    r = 1.0  # Unit circle
    
    z = r * np.exp(1j * theta)
    
    # Compute log(z) = ln(r) + i*theta
    log_z = np.log(r) + 1j * theta
    
    # Also show the principal branch
    log_z_principal = np.log(z)
    
    return theta, log_z, log_z_principal

Visualization: Complete Demonstration

We now create a comprehensive visualization showing:

1. Power series domains: Overlapping circles of convergence

2. Riemann zeta function: Magnitude plot showing analytic continuation

3. Logarithm branches: Spiral structure from multi-valued continuation

4. Error analysis: Comparing series approximations to exact values

# Create comprehensive figure
fig = plt.figure(figsize=(16, 12))

# ==================== Panel 1: Power Series Domains ====================
ax1 = plt.subplot(2, 3, 1)
ax1.set_aspect('equal')

# Draw circles of convergence
circle1 = Circle((0, 0), 1, fill=False, edgecolor='blue', linewidth=2, linestyle='--', label='$|z| < 1$')
circle2 = Circle((0.5, 0), 0.5, fill=False, edgecolor='red', linewidth=2, linestyle='--', label='$|z - 1/2| < 1/2$')
ax1.add_patch(circle1)
ax1.add_patch(circle2)

# Mark centers
ax1.plot(0, 0, 'bo', markersize=8, label='$z_0 = 0$')
ax1.plot(0.5, 0, 'ro', markersize=8, label='$z_1 = 1/2$')
ax1.plot(1, 0, 'kx', markersize=10, label='Singularity at $z=1$')

# Shade overlap region
theta = np.linspace(0, 2*np.pi, 100)
x_overlap = 0.5 * np.cos(theta) + 0.5
y_overlap = 0.5 * np.sin(theta)
mask = x_overlap**2 + y_overlap**2 < 1
ax1.fill(x_overlap[mask], y_overlap[mask], alpha=0.3, color='green', label='Overlap region')

ax1.set_xlim(-1.5, 1.8)
ax1.set_ylim(-1.5, 1.5)
ax1.grid(True, alpha=0.3)
ax1.set_xlabel('Re(z)')
ax1.set_ylabel('Im(z)')
ax1.set_title('Overlapping Domains of Convergence')
ax1.legend(loc='upper left', fontsize=8)

# ==================== Panel 2: Riemann Zeta Magnitude ====================
ax2 = plt.subplot(2, 3, 2)
X, Y, magnitude, zeta_values = plot_zeta_magnitude()

im = ax2.contourf(X, Y, magnitude, levels=20, cmap='viridis')
plt.colorbar(im, ax=ax2, label='$|\\zeta(s)|$')

# Mark the critical line Re(s) = 1/2
ax2.axvline(x=0.5, color='red', linestyle='--', linewidth=1.5, label='Critical line Re(s)=1/2')
# Mark Re(s) = 1 (boundary of series convergence)
ax2.axvline(x=1.0, color='white', linestyle=':', linewidth=1.5, label='Series boundary Re(s)=1')

# Mark some known zeros on critical line
zeros_im = [14.134725, 21.022040, 25.010858, 30.424876]  # First few non-trivial zeros
for zero_im in zeros_im:
    ax2.plot(0.5, zero_im, 'r*', markersize=8)
    ax2.plot(0.5, -zero_im, 'r*', markersize=8)

ax2.set_xlabel('Re(s)')
ax2.set_ylabel('Im(s)')
ax2.set_title('Riemann Zeta Function: Analytic Continuation')
ax2.legend(loc='upper right', fontsize=8)

# ==================== Panel 3: Logarithm Branches ====================
ax3 = plt.subplot(2, 3, 3)
theta, log_z, log_z_principal = demonstrate_logarithm_branches()

ax3.plot(np.real(log_z), np.imag(log_z), 'b-', linewidth=2, label='Multi-valued log(z)')
ax3.plot(np.real(log_z_principal), np.imag(log_z_principal), 'r--', linewidth=2, label='Principal branch')

# Mark branch cuts
ax3.axhline(y=0, color='green', linestyle=':', alpha=0.5, label='Branch cut')
ax3.axhline(y=2*np.pi, color='green', linestyle=':', alpha=0.5)
ax3.axhline(y=-2*np.pi, color='green', linestyle=':', alpha=0.5)

ax3.set_xlabel('Re(log z)')
ax3.set_ylabel('Im(log z)')
ax3.set_title('Complex Logarithm: Branch Structure')
ax3.legend(fontsize=8)
ax3.grid(True, alpha=0.3)

# ==================== Panel 4: Convergence Comparison ====================
ax4 = plt.subplot(2, 3, 4)

# Test points along real axis
x_test = np.linspace(-0.5, 1.5, 200)
z_test = x_test + 0.0j

# Avoid singularity
mask = np.abs(z_test - 1.0) > 0.05
z_safe = z_test[mask]

exact = exact_function(z_safe)
series_0 = power_series_at_origin(z_safe, n_terms=30)
series_half = power_series_at_half(z_safe, n_terms=30)

ax4.plot(np.real(z_safe), np.real(exact), 'k-', linewidth=2, label='Exact: $1/(1-z)$')
ax4.plot(np.real(z_safe), np.real(series_0), 'b--', linewidth=1.5, label='Series at $z_0=0$')
ax4.plot(np.real(z_safe), np.real(series_half), 'r:', linewidth=1.5, label='Series at $z_1=1/2$')

ax4.axvline(x=0, color='blue', alpha=0.3, linestyle='--')
ax4.axvline(x=1, color='gray', alpha=0.3, linestyle='--', label='Singularity')
ax4.set_xlabel('Re(z)')
ax4.set_ylabel('Re(f(z))')
ax4.set_title('Power Series Approximations (Real Part)')
ax4.legend(fontsize=8)
ax4.grid(True, alpha=0.3)
ax4.set_ylim(-5, 5)

# ==================== Panel 5: Error Analysis ====================
ax5 = plt.subplot(2, 3, 5)

error_0 = np.abs(exact - series_0)
error_half = np.abs(exact - series_half)

ax5.semilogy(np.real(z_safe), error_0, 'b-', linewidth=2, label='Error: series at $z_0=0$')
ax5.semilogy(np.real(z_safe), error_half, 'r-', linewidth=2, label='Error: series at $z_1=1/2$')

# Mark radius of convergence boundaries
ax5.axvline(x=0, color='blue', alpha=0.3, linestyle='--', label='$z_0=0$')
ax5.axvline(x=0.5, color='red', alpha=0.3, linestyle='--', label='$z_1=1/2$')
ax5.axvline(x=1.0, color='gray', alpha=0.5, linestyle='--')

ax5.set_xlabel('Re(z)')
ax5.set_ylabel('Absolute Error')
ax5.set_title('Convergence Error Analysis')
ax5.legend(fontsize=8)
ax5.grid(True, alpha=0.3)

# ==================== Panel 6: Phase Portrait ====================
ax6 = plt.subplot(2, 3, 6)

# Create phase portrait of 1/(1-z)
x_phase = np.linspace(-0.5, 2, 300)
y_phase = np.linspace(-1, 1, 300)
X_phase, Y_phase = np.meshgrid(x_phase, y_phase)
Z_phase = X_phase + 1j*Y_phase

# Avoid singularity
f_phase = 1.0 / (1.0 - Z_phase)
phase = np.angle(f_phase)

im = ax6.contourf(X_phase, Y_phase, phase, levels=50, cmap='hsv')
plt.colorbar(im, ax=ax6, label='Phase (radians)')

# Mark singularity and expansion points
ax6.plot(0, 0, 'bo', markersize=10, label='$z_0=0$')
ax6.plot(0.5, 0, 'ro', markersize=10, label='$z_1=1/2$')
ax6.plot(1, 0, 'kx', markersize=12, markeredgewidth=3, label='Pole')

# Draw circles of convergence
circle1_phase = Circle((0, 0), 1, fill=False, edgecolor='blue', linewidth=1.5, linestyle='--')
circle2_phase = Circle((0.5, 0), 0.5, fill=False, edgecolor='red', linewidth=1.5, linestyle='--')
ax6.add_patch(circle1_phase)
ax6.add_patch(circle2_phase)

ax6.set_xlabel('Re(z)')
ax6.set_ylabel('Im(z)')
ax6.set_title('Phase Portrait: $f(z) = 1/(1-z)$')
ax6.legend(fontsize=8, loc='upper left')
ax6.set_aspect('equal')

plt.tight_layout()
plt.savefig('analytic_continuation_analysis.png', dpi=300, bbox_inches='tight')
print("Visualization saved as plot.png")
plt.show()

Visualization saved as plot.png

Key Insights

1. Domain Extension: The overlapping circles demonstrate how power series can be used to extend analytic functions beyond their initial radius of convergence.

2. Uniqueness: Within the overlap region (green shaded area), both power series converge to the same value, illustrating the Identity Theorem.

3. Riemann Zeta: The color map shows that $\zeta(s)$ is well-defined even for $\text{Re}(s) < 1$, despite the series only converging for $\text{Re}(s) > 1$. The red stars mark non-trivial zeros on the critical line.

4. Branch Structure: The logarithm plot reveals that continuing around the origin increases the imaginary part by $2\pi i$, demonstrating multi-valuedness.

5. Convergence Regions: The error analysis shows that each series approximation is accurate within its radius of convergence and diverges beyond it.

6. Phase Continuity: The phase portrait shows smooth variation except at the singularity, confirming analytic structure.

Physical Applications

Analytic continuation appears throughout physics:

• Quantum Field Theory: Wick rotation continues physical time $t$ to imaginary time $it$ for path integral calculations

• Statistical Mechanics: Partition functions are continued from discrete to continuous temperatures

• Scattering Theory: S-matrix elements are continued from physical to unphysical energy values to locate resonances

• String Theory: Veneziano amplitude requires analytic continuation in Mandelstam variables