Countour integral using residue theorem: $ \int_{\gamma} \tanh(z) dz $ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$.

Question

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$.

I want to do this using the residue theorem but I am unsure of how to work out the poles of $\tanh(z)$. I've tried to use the Taylor expansion and writing it in terms of $\cosh(z)$ and $ \sinh(z) $ but I'm getting confused.

Answer 1

We know that the zeroes of $\cosh z$ appear in $Z=i\left(\frac{\pi}{2}+\pi\mathbb{Z}\right)$, since $\cosh z = \cos(iz)$, and they are all simple zeroes. Over $Z$, $\sinh z$ is a regular non-vanishing function (by the relation $\cosh^2 z-\sinh^2 z=1$, if you like). Since $\frac{\pi}{2}>1$, $\tanh z$ has no singularities in the unit disk, so its contour integral over $\gamma$ is just zero by the residue theorem.