Complex Analysis — Definition, Formula & Examples

Complex analysis is the branch of mathematics that studies functions whose inputs and outputs are complex numbers. It extends calculus to the complex plane, revealing powerful results about differentiability, integration, and series that have no direct analogue in real analysis.

Complex analysis is the study of holomorphic (complex-differentiable) functions $f: U \to \mathbb{C}$ , where $U$ is an open subset of $\mathbb{C}$ . A function $f(z)$ is holomorphic at a point $z_0$ if the limit $\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$ exists, where $h \in \mathbb{C}$ . This single condition implies the function is infinitely differentiable and analytic (representable by a convergent power series).

Key Formula

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

Where:

$u(x,y)$ = Real part of f(z)
$v(x,y)$ = Imaginary part of f(z)
$x$ = Real part of z
$y$ = Imaginary part of z

How It Works

You write a complex function as

f(z) = u(x,y) + iv(x,y)

, where

z = x + iy

and

u, v

are real-valued functions. For

f

to be holomorphic,

u

and

v

must satisfy the Cauchy-Riemann equations:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

and

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

. Once a function is known to be holomorphic, you can apply powerful theorems — Cauchy's integral theorem, the residue theorem, and Laurent series — to evaluate integrals, count zeros, and much more. Many results that are difficult or impossible in real analysis become elegant in the complex setting.

Worked Example

Problem: Verify that f(z) = z² is holomorphic by checking the Cauchy-Riemann equations.

Step 1: Write z² in terms of x and y. Since z = x + iy, compute z² and identify the real and imaginary parts.

z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy

Step 2: Identify u and v from the expansion.

u(x,y) = x^2 - y^2, \quad v(x,y) = 2xy

Step 3: Compute the partial derivatives and check both Cauchy-Riemann equations.

\frac{\partial u}{\partial x} = 2x = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -2y = -\frac{\partial v}{\partial x}

Answer: Both Cauchy-Riemann equations are satisfied everywhere, confirming that f(z) = z² is holomorphic on all of ℂ (it is an entire function).

Why It Matters

Complex analysis is essential in electrical engineering (AC circuit analysis and signal processing), fluid dynamics (conformal mappings model ideal fluid flow), and quantum physics. In pure mathematics, the residue theorem lets you evaluate real integrals that resist all standard calculus techniques.

Common Mistakes

Mistake: Assuming complex differentiability works like real differentiability — that a function can be differentiable in one direction but not another.

Correction: Complex differentiability requires the derivative limit to be the same regardless of the direction h approaches 0 in ℂ. This is far more restrictive than real differentiability and is exactly what the Cauchy-Riemann equations encode.