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Just a quick question I've been wondering about. When would I use Cauchy's Integral Formula over the Residue Theorem to solve complex integration problems with poles? To me it seems that Residue theorem is much faster and easier to apply then creating partial fractions for rationals then solving the individual integrals. Any guidance would be greatly appreciated?

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  • $\begingroup$ It's a theoretical tool that is useful for proving many things, even though most of the time it might not be the best formula to compute an integral. In some sense it is kind of similar to Cramer's rule for solving linear system: no one uses it for practical purposes. $\endgroup$ – BigbearZzz Dec 21 '18 at 4:49
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It just so happens that the Residue Theorem encompasses the Cauchy's Integral Formula. If you have a analytic function $f$ in the domain D such that $\Gamma \in D$ and you are asked to evaluate a loop integral of the form $$\oint_\Gamma \frac{f(z)}{(z-z_0)^n}$$ then just use Cauchy's Integral Formula. It turns out its the exact same as the Residue Formula. Note that they both are essentially both looking at the $1/z$ term of the Laurent expansion of a series, and the Residue theorem encompasses everything without doing much preprocessing work.

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  • $\begingroup$ Okay that makes sense. It just seemed easier to apply residue on most things. $\endgroup$ – Safder Dec 22 '18 at 0:01

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