Complex Analysis 34 | Residue theorem [dark version]

TL;DR

Residue theorem converts a closed contour integral into 2πi times a sum of residues at isolated singularities.

Briefing Cornell Notes

Briefing

Residue theorem turns contour integrals in the complex plane into a bookkeeping problem: once a holomorphic function’s isolated singularities are known, every closed contour integral is determined by the residues at those points, weighted by how many times the contour winds around each singularity. In its simplest form, surrounding a single isolated singularity z0 with a small circle gives

∮ f(z) dz = 2πi · Res(f, z0).

Because the value depends only on the winding behavior (not the exact circle radius), the same residue controls the integral for any closed curve that encloses z0 and no other singularities.

The general version extends this from one singularity to finitely many. Start with an open domain D in the complex plane and a holomorphic function f on D except at finitely many isolated singularities z1, …, zn. Take a closed curve γ whose image lies in D and whose interior stays inside D except for those isolated singularities—meaning γ does not loop around any non-isolated singularities. Under these conditions, the residue theorem states that the contour integral is

∮γ f(z) dz = 2πi · Σ_{j=1..n} (wind(γ, zj) · Res(f, zj)).

Here, wind(γ, zj) is the winding number: an integer measuring how many times γ encircles zj, with orientation taken into account. The theorem’s practical punchline is that knowing the residues is enough to compute the entire closed contour integral; the geometry of γ matters only through winding numbers.

The proof strategy relies on Cauchy’s integral theorem and the earlier “keyhole contour” technique. To keep the argument clean, the discussion assumes D is an open disk with the singularities removed (a punctured disk). The curve γ is then expanded slightly so that, around each singularity, a keyhole contour can be inserted. On each keyhole contour, Cauchy’s theorem applies on the region where f is holomorphic, forcing the integral around the modified boundary to vanish. What remains is an expression of the original integral as a sum of integrals around small circles centered at each singularity. Each small-circle integral contributes 2πi times the corresponding residue, and the winding number records how many times the contour effectively wraps each singularity.

The takeaway is both conceptual and computational. Conceptually, residues are the local data that control global contour integrals. Computationally, the theorem reduces many contour-integral problems to (1) identifying isolated singularities, (2) computing residues, and (3) determining winding numbers for the chosen contour. The result also sets up later applications: contour integrals built from residues can be used to evaluate real integrals, which is flagged as the next step in the series.

Cornell Notes

Residue theorem provides a direct formula for closed contour integrals of a holomorphic function with isolated singularities. For a function f holomorphic on a domain D except at finitely many isolated points z1,…,zn, and a closed curve γ lying in D whose interior contains no other non-isolated singularities, the integral is

∮γ f(z) dz = 2πi Σ_{j=1..n} wind(γ, zj) Res(f, zj).

The winding number wind(γ, zj) counts how many times γ encircles each singularity (with orientation). The theorem generalizes the single-singularity case where ∮ f(z) dz equals 2πi times the residue. Proof uses Cauchy’s integral theorem and keyhole contours to rewrite the original integral as a sum of small-circle integrals around each isolated singularity.

What conditions on the domain D, the function f, and the curve γ are required for the residue theorem to apply?

D must be an open (connected) domain in ℂ. The function f must be holomorphic on D except for finitely many isolated singularities z1,…,zn. The closed curve γ must lie in D, and its interior must stay inside D except for those isolated singularities—so γ must not surround any other non-isolated singularities. A common sufficient setup is when D is an open disk with the singularities removed; then the interior condition holds for any curve that stays within that punctured disk.

How does the winding number enter the residue theorem, and what does it measure?

The winding number wind(γ, zj) is an integer that measures how many times the curve γ wraps around the point zj, including the direction (orientation). In the formula, each residue Res(f, zj) is multiplied by wind(γ, zj), so the contour’s geometry affects the integral only through these integer winding counts, not through the specific shape of γ.

Why does the contour integral depend only on residues and winding numbers, not on the exact contour shape?

For a single isolated singularity z0, any sufficiently small circle around z0 gives the same value because the difference between two such contours can be handled using keyhole contour reasoning and Cauchy’s theorem on regions where f is holomorphic. In the multi-singularity case, the same idea lets one deform the contour while accounting for how many times it encircles each isolated singularity; those counts are exactly the winding numbers.

What is the proof idea when D is taken as a punctured disk?

Assume D is an open disk with the singularities removed (D = D~ without the singularities). Enlarge the curve γ slightly so that around each isolated singularity one can insert a keyhole contour. On the resulting boundaries, Cauchy’s integral theorem applies because f is holomorphic on the region enclosed by each keyhole contour. The integral around the modified boundary becomes zero, and the original integral can be expressed as a sum of integrals around small circles centered at each singularity.

How does the single-singularity formula relate to the general residue theorem?

With one isolated singularity z0 inside γ and no others, the sum in the general theorem has only one term. The result becomes ∮γ f(z) dz = 2πi · wind(γ, z0) · Res(f, z0). For a positively oriented simple loop around z0, wind(γ, z0)=1, recovering the familiar ∮ f(z) dz = 2πi Res(f, z0).

Review Questions

State the residue theorem for finitely many isolated singularities and explain the role of the winding number.
What interior condition must the curve γ satisfy to ensure the residue theorem applies?
Outline how keyhole contours and Cauchy’s integral theorem combine to reduce a general contour integral to a sum of small-circle integrals around singularities.

Key Points

1
Residue theorem converts a closed contour integral into 2πi times a sum of residues at isolated singularities.
2
Each residue Res(f, zj) is multiplied by the winding number wind(γ, zj), capturing how many times γ encircles zj.
3
The curve γ must not surround any non-isolated singularities; its interior must lie in the domain except at the isolated singularities.
4
For a single isolated singularity z0, the integral reduces to ∮γ f(z) dz = 2πi · wind(γ, z0) · Res(f, z0).
5
A clean proof strategy uses Cauchy’s integral theorem plus keyhole contours to rewrite the integral as a sum over small circles around each singularity.
6
When the domain is a punctured disk, contour deformation plus keyhole contours make the multi-singularity case essentially a sum of the single-singularity contributions.

Highlights

For isolated singularities, the contour integral is determined entirely by local data (residues) and global wrapping behavior (winding numbers).

The general formula is ∮γ f(z) dz = 2πi Σ wind(γ, zj) Res(f, zj), with finitely many isolated singularities.

Keyhole contours let Cauchy’s integral theorem eliminate parts of the contour integral, leaving only contributions from small circles around singularities.