I'm supposed to examine a function, find the singular part.

$\displaystyle f(z)$ = $\displaystyle \displaystyle \frac{z^2+1}{z(e^{2\pi z} -1)}$

I start with the exponent and do a Taylor expansion.

$\displaystyle \displaystyle e^{2\pi z} -1 $ = $\displaystyle \displaystyle 1 + 2\pi z + 2 \pi^2 z^2 + O(z^3) + ... - 1$

All together.

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

From this I get

$\displaystyle \displaystyle \frac{1}{2\pi z^2} - \frac{1}{2 z} + \frac{1}{2 \pi} - \frac{z}{2} + ...$

Conclusions drawn from this:

$\displaystyle \displaystyle \frac{1}{2\pi z^2} - \frac{1}{2 z}$ - Singular part

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

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

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