Classifying the singularity of $( e^{\sqrt{z}} - e^{-\sqrt z})/\sin \sqrt{z}$ at zero I have a function,
$$ \frac{ e^{\sqrt{z}} - e^{-\sqrt z}}{\sin \sqrt z} $$
Does this function have removable singularities? to me it seems like there is a branch point at $z=0$. Thanks for any inputs.

Let's start with the function $g(w) = (e^w-e^{-w})/\sin w$, which has a removable singularity at $w=0$. The question to address: is $g$ an even function? If it is, then yes, we can plug $\sqrt{z}$ into it and not worry about branches at all: the composition will be holomorphic near zero. (All powers in the Taylor series of an even function at $0$ are even.) 
So: is $g$ even? I guess it is! 
