Types of singularity of function $ f(z)=\frac{e^{iz}}{e^{z}+e^{-z}} $

I am looking for types of singularity of function $$ f(z)=\frac{e^{iz}}{e^{z}+e^{-z}} $$ Since $e^z + e^{−z} = 0$ if and only $e^{2z} = −1$, the singularities of $f$ are exactly the solutions in $\mathbb{C}$ of $e^{2z}= −1$. Write $z = a + ib$, so that $e^{2z} = e^{2a}e^{2bi}$. we get $e^{2a} = 1$ then $a = 0$. We then have $e^{2ib} = −1=e^{i\pi}$ whose solutions are $b = \frac{\pi}{2} + k\pi$ $k\in\mathbb{Z}$. We deduce that the set of solutions is $\{i(\frac{\pi}{2} + k\pi): k\in\mathbb{Z}\}$.

My problem: Why is the nature of these singularities a pole of order 1?

Solution 1:

If $w$ is a zero of $g(z)=e^z+e^{-z}$, then\begin{align}g'(w)&=e^w-e^{-w}\\&=e^w+e^{-w}-2e^{-w}\\&=-2e^{-w}\\&\ne0,\end{align}and therefore every zero of $g$ is a simple zero.