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I'm aware that the inverse $z$-transform can be evaluated using contour integration which leads to the use of Residue Theory as a corollary and I do know of the two definitions. My question is how does one lead to the other being used? It's been many years since I studied the two.

Thanks.

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It's a consequence of Cauchy's Residue Theorem, which can be used to evaluate contour integrals of analytic functions. Take a look at this answer for an example.

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