Complex Analysis: The Theory of Holomorphic Functions

Complex analysis is the study of functions that are complex-differentiable in an open subset of the complex plane $\mathbb{C}$. While it may seem like a straightforward extension of real analysis, the shift from $\mathbb{R}$ to $\mathbb{C}$ introduces a level of rigidity and interconnectedness that is unparalleled. A single derivative in the complex sense implies infinite differentiability and analyticity—a property not shared by real functions.

1. Holomorphic Functions and the Cauchy-Riemann Equations

Let $f:\Omega \to \mathbb{C}$ be a function defined on an open set $\Omega \subseteq \mathbb{C}$. We say $f$ is holomorphic (or complex-differentiable) at $z_0\in \Omega$ if the following limit exists:

$f^{\prime }(z_0)={\lim }_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}$

Unlike the real derivative, $\Delta z$ can approach zero from any direction in the plane. This multidimensional constraint leads to the Cauchy-Riemann (CR) equations. If $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$, then $f$ is holomorphic if and only if:

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Harmonicity

A direct consequence of the CR equations is that the real and imaginary parts of a holomorphic function are harmonic functions, meaning they satisfy Laplace’s equation: $\Delta u=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$ This links complex analysis deeply with potential theory and physics.

2. Complex Integration: The Cauchy-Goursat Theorem

The behavior of holomorphic functions under integration is defined by their local “perfection.” The Cauchy-Goursat Theorem states that if $f$ is holomorphic in a simply connected region $\Omega$, then for any closed contour $\gamma$ in $\Omega$:

${\oint }_{\gamma }f(z)dz=0$

This theorem is the cornerstone of the field. It implies that the integral of a holomorphic function is path-independent, which allows for the definition of an antiderivative $F(z)$ such that $F^{\prime }(z)=f(z)$.

3. Cauchy’s Integral Formula

Perhaps the most surprising result in complex analysis is that the values of a holomorphic function inside a disk are completely determined by its values on the boundary.

Theorem (Cauchy’s Integral Formula): Let $f$ be holomorphic on a disk $D$ and $\gamma$ be its boundary. For any $z_0$ in the interior of $D$:

$f(z_0)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-z_0}dz$

By differentiating under the integral sign, we obtain the formula for higher derivatives: $f^{(n)}(z_0)=\frac{n!}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{(z-z_0{)}^{n+1}}dz$ This proves that if a function is once differentiable in $\mathbb{C}$, it is infinitely differentiable.

4. Power Series: Taylor and Laurent Distributions

Holomorphic functions are locally equivalent to power series.

Taylor Series

If $f$ is holomorphic in a disk $|z-z_0|<R$, it has a unique power series representation: $f(z)={\sum }_{n=0}^{\infty }{a}_n(z-z_0{)}^n,a_n=\frac{f^{(n)}(z_0)}{n!}$

Laurent Series

When a function is holomorphic in an annulus $r<|z-z_0|<R$, we use the Laurent series, which includes negative powers: $f(z)={\sum }_{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n$ The “regular part” consists of $n\ge 0$, and the “principal part” consists of $n<0$.

5. Classification of Singularities

Isolated singularities $z_0$ are categorized by the behavior of the principal part of the Laurent series:

1. Removable Singularity: No principal part ($a_n=0$ for all $n<0$).
2. Pole of order $m$: Principal part is finite ($a_{-m}\ne 0$ and $a_n=0$ for $n<-m$).
3. Essential Singularity: Principal part is infinite.

Casorati-Weierstrass Theorem: Near an essential singularity, a holomorphic function comes arbitrarily close to any complex value. This highlights the chaotic behavior of functions like $e^{1/z}$ near $z=0$.

6. The Residue Theorem

The Residue of $f$ at $z_0$ is the coefficient $a_{-1}$ in its Laurent expansion. The Residue Theorem generalizes Cauchy’s Integral Formula:

${\oint }_{\gamma }f(z)dz=2\pi i{\sum }_{k=1}^n\text{Res}(f,z_k)$

This allows for the evaluation of difficult real integrals by extending them into the complex plane as contours (e.g., using Jordan’s Lemma).

7. Conformal Mappings and Riemann Mapping Theorem

A holomorphic function with $f^{\prime }(z)\ne 0$ is a conformal mapping; it preserves angles and the orientation of curves.

Riemann Mapping Theorem: Any simply connected open subset $\Omega ⊊\mathbb{C}$ can be mapped biholomorphically to the open unit disk $\mathbb{D}$. This is a profound result connecting topology and analysis.

8. Analytic Continuation

If two holomorphic functions $f$ and $g$ agree on a set with an accumulation point, they are identical everywhere on their connected domain. This allows for analytic continuation, where we extend the definition of a function beyond its original radius of convergence. A famous example is the Riemann Zeta Function $\zeta (s)$, originally defined for $\text{Re}(s)>1$, which is continued to the rest of the plane (except $s=1$).

Python Visualization: Conformal Mapping

The following script visualizes the transformation of a Cartesian grid under the mapping $f(z)=z^2$. Note how the orthogonality of the grid lines is preserved (except at the origin).

import numpy as np
import matplotlib.pyplot as plt

def plot_conformal():
    # Create a grid in the complex plane
    u = np.linspace(-2, 2, 40)
    v = np.linspace(-2, 2, 40)
    U, V = np.meshgrid(u, v)
    Z = U + 1j*V

    # Define the mapping f(z) = z^2
    W = Z**2

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

    # Plot domain
    for i in range(len(u)):
        ax1.plot(Z[i, :].real, Z[i, :].imag, color='blue', alpha=0.3)
        ax1.plot(Z[:, i].real, Z[:, i].imag, color='red', alpha=0.3)
    ax1.set_title("Domain: z-plane")
    ax1.grid(True)

    # Plot range
    for i in range(len(u)):
        ax2.plot(W[i, :].real, W[i, :].imag, color='blue', alpha=0.3)
        ax2.plot(W[:, i].real, W[:, i].imag, color='red', alpha=0.3)
    ax2.set_title("Range: w = z²")
    ax2.grid(True)

    plt.tight_layout()
    plt.show()

if __name__ == "__main__":
    plot_conformal()