Cauchy's Residue Theorem

If $𝑓 (𝑧)$ is analytic over a closed region $𝐷 \subseteq ℂ$ except at isolated points, i.e. it is holomorphic for $𝐷 ∖ {𝑧 1, 𝑧 2, \dots, 𝑧 𝑀}$ , then the contour integral

\oint 𝜕 𝐷 𝑓 (𝑧) 𝑑 𝑧 = 2 𝜋 𝑖 𝑀 \sum 𝑚 = 1 R e s [𝑓, 𝑧 𝑚]

where $R e s [𝑓, 𝑧 𝑚]$ is the Residue of the singular point $𝑧 𝑚$ which is relatively easy to calculate.

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