Residue

A residue of a meromorphic function $𝑓$ at a Singular point $𝑧 0$ is the value of the coëfficient $𝐴 - 1$ of the Laurent series about $𝑧 0$ .

R e s [𝑓, 𝑧 0] = 𝐴 - 1

Residues themselves can be used to calculate a contour integral using Cauchy's Residue Theorem.

Calculating residues

Cauchy's residue formula¹ states that if $𝑓 (𝑧)$ has a pole of order $𝑘$ at $𝑧 = 𝑧 0$ , then

R e s [𝑓, 𝑧 0] = 1 (𝑘 - 1)! l i m 𝑧 \to 𝑧 0 𝑑 𝑘 - 1 𝑑 𝑧 𝑘 - 1 [(𝑧 - 𝑧 0) 𝑘 𝑓 (𝑧)]

In many cases, using the Laurent series is easier than using this formula.

Often L'Hôpital's Rule is involved in calculating residues.

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