Compute $\oint_{\gamma}f(z)dz$, where $f(z)=\frac {ze^{\pi z}}{z^4-16}+ze^{\frac \pi z}$ where $\gamma:9x^2+y^2=9$ Using the residue theorem.
I don't know how to start this, first I thought about parametrizing this curve as: $x=\frac 13\cos\theta$ and $y=\sin\theta$ but I don't know how this would help me.. any ideas?
The curve $\gamma$ (directed counterclockwise) is an ellipse having the poles $\pm2i$ of the first part in its interior, but not the poles $\pm2$. For the first part of the integrand it is therefore sufficient to compute the residues at $\pm2i$, using standard computational methods (no computing of line integrals).
The second part of the integrand has an essential singularity at $0$, but the Laurent expansion $$z\,e^{\pi/z}=\sum_{k=0}^\infty {\pi^k\over k!}\,z^{1-k}$$ is convergent in the full punctured $z$-plane. The residue at $0$ can then be immediately read off.