Gunni
Feb1-07, 02:33 PM
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.
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.