I am supposed to compute the singularities and their kind of $f$ in $\mathbb{C}$ for the following functions, furthermore I shall compute $\int_{|z|=4}f(z)dz$:
a) $\displaystyle f(z)=\frac{\sin(z)}{e^z-e^{\pi}}$
b) $\displaystyle f(z)=\sin(e^\frac{1}{z})$
I think for b) we have an essential singularity at zero. For a) I thought it must be a pole at $\pi$ but it doesn't work out because of the $\sin$. can someone please help? I guess for computing the integral the residue's theorem will be a good choice.
Yes, $z=0$ is the essential singularity for (b).
For (a), let $\theta=z-\pi$, we have
$$\frac{\sin z}{e^z-e^\pi}=-\frac1{e^\pi}\cdot\frac{\sin \theta}{e^\theta-1}$$
So it is clear that $\theta=0$ is a removable singularity, hence it is analytic at $z=\pi$. But there are infinitely many simple poles at $\theta=2n\pi i, n\in\mathbb Z\land n\neq 0$, which is equivalent to
$$z=\pi+ 2n\pi i,~~ n\in\mathbb Z\land n\neq 0$$