Zeros, Singularities, Residues: Residue Theorem
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Theorem (Cauchy's Residue Theorem): If $C$ is a simple closed positively oriented contour and $f(z)$ is analytic on and inside $C$ except at the finite number of points $z_{1}$, $z_{2}$, $\cdots$, $z_{n}$ inside $C$,
then

MATH

Examaple: Evaluate MATH where $C$ is the unit circle $\vert z \vert = 1$ oriented positively.
The function MATH has only one singular point $z = 0$ inside the circle MATH. The residue of $f(z)$ at $z = 0$ is equal to $-1$. Therefore, by Cauchy's Residue theorem,

MATH

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