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

  • Thread starter Gunni
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Homework Statement


Show that

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

for all a such that [tex] - \pi < a < \pi [/tex]

Homework Equations


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

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.
 

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