When would I use Cauchy's Integral Formula over Residue

Question

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?

Answer 1

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.