Complex Analysis 34 | Residue theorem

TL;DR

For a holomorphic function with an isolated singularity z0, the contour integral around any sufficiently small circle enclosing only z0 equals 2πi times Res(f,z0).

Briefing Cornell Notes

Briefing

Residue theorem turns contour integrals of holomorphic functions with isolated singularities into a bookkeeping problem: the integral around a closed curve is determined entirely by the residues at those singularities, weighted by how many times the curve winds around each one. In its simplest form, if a holomorphic function f is defined on a domain D except for an isolated singularity z0, then the integral of f around any small positively oriented circle enclosing z0 equals 2πi times the residue of f at z0. Because the keyhole-contour argument shows the exact circle radius doesn’t matter (as long as z0 is the only singularity inside), the same value holds for any such loop.

The general residue theorem extends this from one singularity to many. Suppose f is holomorphic on an open domain D (connected) except for 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 surround any non-isolated singularities. Under these conditions, the contour integral over γ is

∮γ f(z) dz = 2πi · Σ(j=1 to n) windingNumber(γ, zj) · Res(f, zj).

The winding number captures the geometric fact that γ may loop around a singularity multiple times; if it winds once, the residue contributes once, and if it winds k times, that residue contributes k times. This is the core practical payoff: once the residues are known, the contour integral follows without directly integrating along γ.

The proof strategy uses Cauchy’s integral theorem in a controlled way. To keep things simple, the discussion assumes D is an open disk with the singularities removed (a punctured disk). The curve γ is chosen so its interior lies within that punctured disk. The key idea is to enlarge γ slightly and, around each isolated singularity, insert a small keyhole contour. On each keyhole contour, Cauchy’s theorem applies to the holomorphic part, forcing the relevant integrals to cancel in a way that leaves only contributions from small circles around the singularities. Those remaining pieces are exactly the 2πi times residue terms, and the winding numbers account for how γ relates to each punctured circle.

By the end, the residue theorem is presented as a powerful generalization of Cauchy’s theorem: it converts complex contour integration into residue computation. The formula is positioned as a tool for later applications, including using contour integrals to evaluate real integrals.

Cornell Notes

Residue theorem provides a direct formula for contour integrals of holomorphic functions with isolated singularities. If f is holomorphic on an open connected domain D except for finitely many isolated singularities z1,…,zn, then for any closed curve γ lying in D whose interior contains only those isolated singularities, the integral satisfies ∮γ f(z) dz = 2πi Σj windingNumber(γ,zj)·Res(f,zj). The key geometric input is the winding number, which counts how many times γ wraps around each singularity. The proof uses Cauchy’s integral theorem by decomposing the region with keyhole contours around each singularity, reducing the integral to a sum of small-circle contributions determined by residues. This makes contour integration largely a matter of computing residues.

Why does the integral around a small circle depend only on the residue and not on the circle’s radius?

For an isolated singularity z0, choose ε small enough that the punctured disk contains no other singularities. The keyhole-contour argument implies that any two circles enclosing z0 (and no other singularities) can be deformed into each other within the domain where f is holomorphic. By Cauchy-type reasoning, the contour integral stays the same. The value is therefore fixed as 2πi·Res(f,z0), regardless of the specific circle radius, as long as the circle encloses only z0.

What role does the winding number play in the residue theorem?

The winding number windingNumber(γ,zj) measures how many times the closed curve γ encircles the singularity zj (with orientation). If γ loops once around zj, that residue contributes once; if it loops k times, the residue contributes k times. This is why the general formula weights each residue by the winding number rather than treating all singularities equally.

What conditions must the curve γ satisfy for the residue theorem to apply?

The curve γ must lie in the domain D, and its interior must not cross any non-isolated singularities. Equivalently, interior(γ) ∪ {isolated singularities} must be contained in D. The transcript notes that this can be ensured, for example, when D is an open disk with the isolated singularities removed, because then the interior of any filled-in loop automatically stays within D except at the punctures.

How does the proof use Cauchy’s integral theorem and keyhole contours?

In the simplified setting where D is a punctured open disk, the curve γ is enlarged so that around each isolated singularity zj one can insert a small keyhole contour. On the remaining region, f is holomorphic, so Cauchy’s integral theorem makes the integral over the boundary of that holomorphic region vanish. What remains are integrals over small circles around each zj, which evaluate to 2πi·Res(f,zj). The relationship between γ and those small circles produces the winding-number factors.

How does the residue theorem generalize the single-singularity case?

With one isolated singularity z0, the integral is 2πi·Res(f,z0). With multiple isolated singularities z1,…,zn, the theorem sums the contributions from each singularity. Each residue Res(f,zj) is multiplied by how many times γ winds around zj, then all weighted residues are added and multiplied by 2πi.

Review Questions

In the residue theorem formula, what geometric quantity determines how strongly each residue contributes to the contour integral?
What does it mean for the interior of γ to avoid non-isolated singularities, and why is that restriction necessary?
Outline the proof idea: how do keyhole contours and Cauchy’s integral theorem combine to reduce the integral to residue terms?

Key Points

1
For a holomorphic function with an isolated singularity z0, the contour integral around any sufficiently small circle enclosing only z0 equals 2πi times Res(f,z0).
2
The keyhole-contour deformation argument shows the circle’s radius doesn’t matter as long as no other singularities enter the contour.
3
For finitely many isolated singularities z1,…,zn, the residue theorem gives ∮γ f(z)dz = 2πi Σj windingNumber(γ,zj)·Res(f,zj).
4
The winding number counts how many times γ wraps around each singularity and weights that singularity’s residue contribution accordingly.
5
The curve γ must lie in the domain D, and interior(γ) must stay inside D except for the isolated singularities.
6
A simplified proof uses Cauchy’s integral theorem on regions where f is holomorphic, inserting keyhole contours around each isolated singularity so only residue contributions survive.

Highlights

Residue theorem converts contour integration into a sum of residues weighted by winding numbers.

Any circle enclosing only one isolated singularity gives the same 2πi·Res(f,z0) value, independent of the circle’s radius.

For multiple singularities, each residue contributes according to how many times the contour winds around it.

The proof idea relies on enlarging γ and inserting keyhole contours so Cauchy’s theorem kills the holomorphic remainder.