REFERENCE SHEET

Laurent series, poles & residues

The coefficient of (z−a)⁻¹ records the local contribution to a contour integral.

Before this Contour integration & Cauchy formulas

∮γ f(z) dz = 2πi Σa Res(f,a) (positive simple boundary)

Definitions

Laurent expansion

On an annulus of holomorphicity about a, f has a unique convergent expansion Σₙ∈ℤ cₙ(z−a)ⁿ.

Isolated singularity

Removable if all negative Laurent coefficients vanish; a pole if finitely many are nonzero; essential if infinitely many are nonzero.

Residue

Res(f,a)=c₋₁. At a simple pole, it is limz→a (z−a)f(z).

Key theorems

Residue theorem

If f is holomorphic on a neighborhood of a positively oriented piecewise smooth simple closed contour and its interior except for finitely many isolated singularities inside, its integral is 2πi times their residue sum.

Simple quotient pole

If g,h are holomorphic near a, h(a)=0, h′(a)≠0, and g(a)≠0, then Res(g/h,a)=g(a)/h′(a).

Removable singularity theorem

A holomorphic function on a punctured neighborhood of a that is bounded near a extends holomorphically across a.

Notable applications

Real integrals

Suitable complex contours can evaluate real integrals; estimates must justify any arc contributions being discarded.

Coefficient extraction

Contour integrals recover power-series coefficients and support generating-function methods.

Work it out

Try before revealing

01 / Compute

Find Res(1/(z²+1),i).

Hint

The denominator derivative is 2z.

Solution

The simple-pole formula gives 1/(2i).

02 / Classify

Classify the singularity of sin(z)/z at 0 and give its residue.