LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-27

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An indented arc around a simple singularity contributes the expected residue fraction

Statement

Let $f$ have a simple pole at $a$ , let $0 < ε < R$ , and for arbitrary real angles $α, β$ let $γ_{ε} (t) = a + ε e^{i ((1 - t) α + t β)} (0 \leq t \leq 1)$ be the circular arc oriented from angle $α$ to angle $β$ inside $0 < ∣ z - a ∣ < R$ . Then

$\lim_{ε ↓ 0} \int_{γ_{ε}} f (z) d z = i (β - α) Res (f, a) .$

In particular, an upper indentation from left to right contributes $- i π Res (f, a)$ and a lower indentation contributes $+ i π Res (f, a)$ .

Facts & Assumptions

Given: A simple pole of $f$ at $a$ and the oriented arc $γ_{ε} (t) = a + ε e^{i ((1 - t) α + t β)}$ .

[L1]

At a simple pole, the Laurent principal part is $c_{- 1} / (z - a)$ , where $c_{- 1} = Res (f, a)$ ; hence $f (z) = c_{- 1} / (z - a) + h (z)$ with $h$ holomorphic near $a$ (Simple poles, The residue of an isolated singularity).

Proof

technique · direct

1.1L1algebra

Write $f (z) = c_{- 1} (z - a)^{- 1} + h (z)$ as in [L1]. Along the arc, put $θ (t) = (1 - t) α + t β$ . Then $z - a = ε e^{i θ (t)}$ and $d z = i (β - α) ε e^{i θ (t)} d t$ , so $\int_{γ_{ε}} \frac{c_{- 1}}{z - a} d z = i (β - α) c_{- 1} = i (β - α) Res (f, a) .$

2.1step 1.1

The holomorphic function $h$ is bounded on a small closed disc around $a$ , say by $M$ . Hence $∣ \int_{γ_{ε}} h (z) d z ∣ \leq M ε ∣ β - α ∣,$ which tends to $0$ with $ε$ .

3.1step 1.1step 2.1∎

Adding the two parts from steps 1.1 and 2.1 proves the limit formula. The two indentation special cases are the choices $(α, β) = (π, 0)$ and $(α, β) = (π, 2 π)$ .

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Howell and J. Mathews, Complex Analysis, Ch. 8 §8.5 (standard reference, not scraped)