Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Feb 1, 2007 #1
    1. The problem statement, all variables and given/known data
    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]

    2. Relevant 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.

    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.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?