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The residue theorem for a null-homologous cycle

Statement

Let $Ω \subseteq C$ be open, let $f$ be meromorphic on $Ω$ with pole set $S$ , and let $Γ$ be admissible for the residue theorem in $Ω$ . Then

$\int_{Γ} f (z) d z = 2 π i \sum_{a \in S} n (Γ, a) Res (f, a),$

where only finitely many terms are nonzero.

Facts & Assumptions

Given: An open set $Ω$ , a meromorphic function $f$ on $Ω$ with pole set $S$ , and an admissible cycle $Γ$ in $Ω$ .

[L1]

Only finitely many poles have nonzero index with respect to $Γ$ (Only finitely many singularities contribute to the residue sum of an admissible cycle).

[L2]

For a sufficiently small positively oriented circle $C (a, r)$ around an isolated singularity $a$ , the integral of $f$ on that circle is $2 π i Res (f, a)$ (The residue is the normalized small-circle integral).

[L3]

If two cycles are homologous in an open set on which a function is holomorphic, then their contour integrals agree (Holomorphic integrals agree on homologous cycles).

[L4]

A positively oriented circle around $a$ has index $1$ inside and $0$ outside (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

Proof

technique · direct

1.1givenL1choose

By [L1], the set $A : = {a \in S : n (Γ, a) \neq 0}$ is finite. For each $a \in A$ choose $r_{a} > 0$ so small that the closed discs $\overline{D (a, r_{a})}$ are pairwise disjoint, lie in $Ω$ , meet no pole other than $a$ , and are disjoint from $Γ^{*}$ . Let $C_{a}$ be the positively oriented circle $∣ ζ - a ∣ = r_{a}$ .

2.1L4

Put [step 1.1, L4] $Δ : = \sum_{a \in A} n (Γ, a) C_{a} .$ For every $p \in C ∖ (Ω ∖ A)$ one has $n (Γ, p) = n (Δ, p)$ . Indeed, if $p \notin Ω$ then admissibility makes $n (Γ, p) = 0$ , and every $C_{a}$ lies in $Ω$ , so gives $n (Δ, p) = 0$ as well. If $p = a \in A$ , then [L4] gives $n (C_{a}, a) = 1$ and $n (C_{b}, a) = 0$ for $b \neq a$ , so $n (Δ, a) = n (Γ, a)$ . Therefore $Γ$ and $Δ$ are homologous in $Ω ∖ A$ .

3.1step 2.1L3

The function $f$ is holomorphic on $Ω ∖ A$ , so [L3] applied to step 2.1 yields $\int_{Γ} f (z) d z = \int_{Δ} f (z) d z = \sum_{a \in A} n (Γ, a) \int_{C_{a}} f (z) d z .$

4.1step 3.1L2∎

Each $C_{a}$ encloses only the pole $a$ , so [L2] gives $\int_{C_{a}} f (z) d z = 2 π i Res (f, a) .$ Substituting this into step 3.1 proves the displayed formula. Since the set $A$ is finite, the residue sum has only finitely many nonzero terms.

The residue theorem for a null-homologous cycle

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources