I think my problem below can be solved by finding the Laurent series of $f(z) = \ln(1+\exp(z))$ about its points of singularities. (Any better suggestion is more than welcome!) How might I find such a series?
I think that the function $f(z) = \ln(1+\exp(z))$ has poles at $z=i\pi (2n-1)$ for integer $n$ but I cannot prove that it is a pole because I cannot find an $m$ such that $(z-i\pi (2n-1))^mf(z)$ is finite. Could someone please help me out?
Added: For this question we are taking the principal branch of the function.
By the way, the available options are
A) removable singularities, (Don't think so...)
B) poles, (Possible, but I can't find a suitable $m$, as mentioned)
C) essential singularities, (Possible, but I don't know how to show this either...)
D) non-isolated singularities (Not this one)
Some thoughts: I have thought of maybe obtaining the Laurent series about these $z$ values, but I don't know how to do that...