# Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$

at $z_0=0$?

I don't see an easy way to represent it using Laurent series, neither I don't see how I can find the limit of the function with $z\to 0$.

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Look at $e^{1/z}$ around $z = 0$. This function approaches every point in the complex plane (take the limit along the real axis, positive or negative or along the imaginary axis). Dividing by and adding a meromorphic function does not change this, so there is no singularity that can be characterized in the normal way with Laurent series, residues, etc. This is called an essential singularity.
That's clear, I have the same result, that made me more confident I was thinking in a correct way. But that means that I can't get use of the Residue theory for integration to find a path integral where the path is a circle with a middle point $z_0=0$? – Dmitry Kazakov Jul 19 '14 at 11:38
It is essentially because the part $e^{\frac{1}{z}}=\sum_{n=0}^{\infty}\frac{\frac{1}{z^n}}{n!}=\sum_{n=0}^{\infty}\frac{1}{z^n\cdot n!}$ attains infinity times often addends in the form $\frac{a_k}{z^k}$ (an the other parts are holomorphic or liftable in $0$, in a neighborhood of $0$) so the principal part has infinity many addeds. Thus $z_0=0$ as an essentially singularity.