Compute \[ \int_{S^1} {2 \sinh(z) \over z^n}\dz .\]
Solution
Write \(f(z) = 2\sinh(z) = e^{z} - e^{-z}\) and apply the generalized Cauchy formula: \[ f^{(n-1)}(0) &= {(n-1)! \over 2\pi i} \int_{S^1} {f(z) \over (z-0)^n}\dz \\ \\ \implies \int_{S^1}{f(z)\over z^n}\dz &= {2\pi i \over (n-1)!} f^{(n-1)}(0) \\ &= {2\pi i\over (n-1)!} 2\cdot \qty{e^z - (-1)^{n-1} e^z \over 2}\evalfrom_{z=0} \\ &= {2\pi i\over (n-1)!} 2\cdot \qty{1 + (-1)^{n} \over 2} \\ &= \begin{cases} {4\pi i \over (n-1)!} & n \text{ even } \\ 0 & n \text{ odd }. \end{cases} .\]