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.

A related technique. Here is a start.
$$ \frac{ e^{\sqrt{z}} - e^{-\sqrt z}}{\sin \sqrt z}= \frac{(1+\sqrt{z}-\dots)-(1-\sqrt{z}+\dots)}{\sqrt{z}-\frac{\sqrt{z}}{3!}+\dots}. $$
Can you see what's going on?
