I'm supposed to examine a function, find the singular part.
f(z) = \displaystyle \frac{z^2+1}{z(e^{2\pi z} -1)}

I start with the exponent and do a Taylor expansion.
\displaystyle e^{2\pi z} -1 = \displaystyle 1 + 2\pi z + 2 \pi^2 z^2 + O(z^3) + ... - 1

All together.
\displaystyle \frac {z^2 + 1}{z(2\pi z + 2 \pi^2 z^2 + O(z^3) )} = \displaystyle \frac {z^2 + 1}{2\pi z^2}(1 + \pi z + O(z^2) )}

From this I get
\displaystyle \frac{1}{2\pi z^2} - \frac{1}{2 z} + \frac{1}{2 \pi} - \frac{z}{2} + ...

Conclusions drawn from this:
\displaystyle \frac{1}{2\pi z^2} - \frac{1}{2 z} - Singular part

\displaystyle \frac{1}{2 \pi} - ???

\displaystyle \frac{z}{2} + ... - Real part ???

If anyone would enlighten me on what information I can get from a series, and it's parts, like this?