# A series for sin az / sin pi z in complex analysis

1. Feb 1, 2007

### Gunni

1. The problem statement, all variables and given/known data
Show that

$$\frac{\sin (az)}{\sin (\pi z)} = \frac{2}{\pi} \sum_{n=1}^{+\infty} (-1)^n \frac{n \sin (an)}{z^2 - n^2}$$

for all a such that $$- \pi < a < \pi$$

2. Relevant equations
None really, we have similar expansions for $$\pi cot (\pi z)$$ and $$\pi / \sin (\pi z)$$, this is an excersize in using Mittag-Leffler's theorem.

3. The attempt at a solution
My problem is that I can't show the series is uniformly convergent on every compact subset of C. Once that's done I've got a solution for every rational multiple of pi, which I think can be extended to all real a with a continuity argument. Any thoughts on the convergence problem? It's driving me mad.