Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26
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The residue is the normalized small-circle integral

Statement

Let f have an isolated singularity at a, and suppose f is holomorphic on 0<∣z−a∣<R. For every r with 0<r<R,

Res⁡(f,a)=12πi∫∣ζ−a∣=rf(ζ) dζ.

Facts & Assumptions

Given: An isolated singularity of f at a and a circle ∣ζ−a∣=r inside the punctured neighbourhood.

[L1]

The residue is the coefficient c−1 in the Laurent expansion (The residue of an isolated singularity).

[L2]

Laurent coefficients are given by the contour integrals (2πi)−1∫f(ζ)(ζ−a)−n−1 dζ (Laurent coefficients are given by contour integrals and are unique).

Proof

technique · direct
1.1

Applying [L2] with n=−1 gives the coefficient formula c−1=12πi∫∣ζ−a∣=rf(ζ)(ζ−a)0 dζ=12πi∫∣ζ−a∣=rf(ζ) dζ.

L2algebra
2.1

By [L1], the coefficient c−1 is exactly Res⁡(f,a).

step 1.1L1∎

Depends on

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