Complex Analysis

Complex Analysis is the study of functions of a complex variable $z = x + iy$. It reveals a profound rigidity: if a complex function is differentiable once, it is differentiable infinitely many times.

Holomorphic Functions

A function $f(z) = u(x, y) + i v(x, y)$ is holomorphic if it is complex-differentiable. This requires the Cauchy-Riemann Equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

def check_cauchy_riemann(f, z, h=1e-7):
  # Numerical gradient check
  dz_x = f(z + h) - f(z)
  dz_y = f(z + h*1j) - f(z)
  
  df_dx = dz_x / h
  df_dy = dz_y / (h*1j)
  
  # In a holomorphic function, df/dz should be the same 
  # regardless of the direction of the limit.
  is_holomorphic = abs(df_dx - df_dy) < 1e-5
  return is_holomorphic, df_dx, df_dy

# Test f(z) = z^2 (Holomorphic)
f1 = lambda z: z**2
print(f"z^2 at 1+i: {check_cauchy_riemann(f1, 1+1j)}")

# Test f(z) = conj(z) (Non-holomorphic)
f2 = lambda z: z.conjugate()
print(f"conj(z) at 1+i: {check_cauchy_riemann(f2, 1+1j)}")

Complex Integration and Cauchy’s Theorem

If $f$ is holomorphic in a simply connected domain, the line integral along any closed path $\gamma$ is zero: $\oint_\gamma f(z) dz = 0$

However, if the domain has “holes” (singularities), we apply the Residue Theorem: $\oint_\gamma f(z) dz = 2\pi i \sum \text{Res}(f, z_k)$

Conformal Mappings

Holomorphic functions are conformal maps, meaning they preserve angles between curves. This property is vital in fluid dynamics and electromagnetism.

import numpy as np

def transform_grid(f, points):
  return [f(p) for p in points]

# Grid of points
grid = [x + y*1j for x in np.linspace(-1, 1, 5) for y in np.linspace(-1, 1, 5)]

# Exponential map: mapping lines to circles/rays
f_exp = lambda z: np.exp(z)
transformed = transform_grid(f_exp, grid)

print(f"First 5 Original: {grid[:5]}")
print(f"First 5 Transformed: {[f'{t:.2f}' for t in transformed[:5]]}")

Laurent Series and Singularities

Functions can be expanded into Laurent series near singularities: $f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$

• Removable singularity: No negative power terms.
• Pole of order m: Finite negative terms up to $n = -m$.
• Essential singularity: Infinitely many negative power terms.