Residue Theorem — Definition, Formula & Examples

The Residue Theorem is a powerful result in complex analysis that lets you compute a contour integral around a closed curve by summing up the residues of the function at its isolated singularities inside the curve.

Let $f$ be analytic on and inside a simple closed positively oriented contour $C$ , except at finitely many isolated singularities $z_1, z_2, \ldots, z_n$ interior to $C$ . Then $\oint_C f(z)\,dz = 2\pi i \sum_{k=1}^{n} \operatorname{Res}(f, z_k)$ , where $\operatorname{Res}(f, z_k)$ is the residue of $f$ at $z_k$ , defined as the coefficient $a_{-1}$ in the Laurent series expansion of $f$ about $z_k$ .

Key Formula

\oint_C f(z)\,dz = 2\pi i \sum_{k=1}^{n} \operatorname{Res}(f, z_k)

Where:

$C$ = A simple closed positively oriented contour in the complex plane
$f(z)$ = A function analytic on and inside C except at isolated singularities
$z_k$ = The isolated singularities of f inside C
$\operatorname{Res}(f, z_k)$ = The residue of f at the singularity z_k

How It Works

To apply the Residue Theorem, first identify all isolated singularities of

f(z)

that lie inside your contour

C

. For each singularity, compute the residue — for a simple pole at

z_0

, the residue is

\lim_{z \to z_0}(z - z_0)f(z)

. Multiply the sum of all residues by

2\pi i

to get the value of the integral. The theorem reduces a potentially difficult integration problem to an algebraic computation at a handful of points.

Worked Example

Problem: Evaluate

\oint_C \frac{1}{z(z-2)}\,dz

where

C

is the circle

|z| = 1

, traversed counterclockwise.

Identify singularities: The function has simple poles at

z = 0

and

z = 2

. Only

z = 0

lies inside

|z| = 1

Compute the residue at z = 0: Since

z = 0

is a simple pole, use the limit formula.

\operatorname{Res}(f, 0) = \lim_{z \to 0} z \cdot \frac{1}{z(z-2)} = \lim_{z \to 0} \frac{1}{z-2} = -\frac{1}{2}

Apply the Residue Theorem: Multiply by

2\pi i

\oint_C \frac{1}{z(z-2)}\,dz = 2\pi i \left(-\frac{1}{2}\right) = -\pi i

Answer:

\oint_C \frac{1}{z(z-2)}\,dz = -\pi i

Why It Matters

The Residue Theorem is essential in physics and engineering for evaluating real improper integrals, inverse Laplace transforms, and summing series. In courses on complex analysis and mathematical methods, it is one of the most frequently tested tools.

Common Mistakes

Mistake: Including residues at singularities that lie outside or on the contour.

Correction: Only sum residues at singularities strictly inside the contour. Singularities on the boundary require separate treatment (e.g., indentation arguments).