Holomorphic Function — Definition, Formula & Examples

A holomorphic function is a complex-valued function of a complex variable that is differentiable at every point in its domain. This complex differentiability is a much stronger condition than real differentiability and forces the function to be infinitely differentiable and representable by a power series.

A function $f: U \to \mathbb{C}$ , where $U \subseteq \mathbb{C}$ is open, is holomorphic on $U$ if the limit $f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$ exists for every $z_0 \in U$ , where $h \in \mathbb{C}$ . Equivalently, writing $f(x + iy) = u(x,y) + iv(x,y)$ , the function $f$ is holomorphic if and only if $u$ and $v$ have continuous partial derivatives satisfying the Cauchy-Riemann equations.

Key Formula

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

Where:

$u(x,y)$ = Real part of f(z) = u + iv
$v(x,y)$ = Imaginary part of f(z) = u + iv
$x, y$ = Real and imaginary parts of z = x + iy

How It Works

To check whether a function

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

is holomorphic, you verify 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}

. If both equations hold and the partial derivatives are continuous, the function is holomorphic. A key consequence is that both

u

and

v

are harmonic, meaning they satisfy the Laplacian

\nabla^2 u = 0

and

\nabla^2 v = 0

. This links holomorphic functions directly to solutions of Laplace's equation in two dimensions.

Worked Example

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

Expand in terms of x and y: Write z = x + iy, so f(z) = (x + iy)² = x² - y² + 2ixy. Thus u(x,y) = x² - y² and v(x,y) = 2xy.

f(z) = (x^2 - y^2) + i(2xy)

Compute partial derivatives: Find all four first partial derivatives of u and v.

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

Verify Cauchy-Riemann equations: Check both conditions: u_x = 2x = v_y and u_y = -2y = -v_x. Both hold everywhere.

2x = 2x \quad \checkmark \qquad -2y = -(2y) \quad \checkmark

Answer: Since the Cauchy-Riemann equations hold for all (x, y) and the partial derivatives are continuous, f(z) = z² is holomorphic on all of ℂ (i.e., it is entire).

Why It Matters

Holomorphic functions are central to complex analysis and appear throughout physics and engineering — fluid dynamics models incompressible, irrotational flow using holomorphic functions, and conformal mappings (which are holomorphic with nonzero derivative) solve boundary-value problems in electrostatics. The theory also underpins contour integration techniques used to evaluate difficult real integrals.

Common Mistakes

Mistake: Assuming that a function with continuous partial derivatives in x and y is automatically holomorphic.

Correction: Real smoothness is not enough. The partial derivatives must also satisfy the Cauchy-Riemann equations. For example, f(z) = z̄ = x − iy has smooth components but fails the Cauchy-Riemann equations and is not holomorphic.