I have been trying to determine the singularities of $\frac{z}{\cos z}$:
I tried this problem by finding the series $\frac{1}{\cos (z)}$ separately, and then multiplying like: $$z*\sum_{n=0}^{\infty} a_n z^{2n}$$.
Since we are not concerned with the coefficients, $a_n$, this tells me that there is no singularity at all, because we only have positive powers of $z$. So what types of singularities are these/this?
Other resources indicate that my expansion of 1/ cos z is correct: Find series expansion of 1/cosx
But I'm also other answers different from mine:
1) In this textbook on page 204: An Introduction to Complex Analysis by Agarwal, it says that the answer is $\frac{\pi}{2} + \pi$
Which is correct? My answer? or $\frac{\pi}{2} + \pi$ in Agarwal?
Hint: Do the following: 1. Find the zeros of $\cos z.$ 2. Show that these are simple zeros of $\cos z.$ 3. Show that if $f,g$ are holomorphic near $a,$ $g$ has a simple zero at $a,$ and $f(a)\ne 0,$ then $f/g$ has a simple pole at $a.$