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Jacobi Theta Function modularity translation quick q

  1. Dec 3, 2016 #1
    1. The problem statement, all variables and given/known data

    I have the Jacobi theta series: ##\theta^{m}(\tau) = \sum\limits^{\infty}_{n=0} r_{m}(\tau) q^{n} ##,

    where ##q^{n} = e^{2\pi i n \tau} ## and I want to show that ##\theta^{m}(\tau + 1) = \theta^{m}(\tau) ##

    (dont think its needed but) where ##r_{m} = ## number of ways of writing ##m## as the sum of ##n## squares.

    2. Relevant equations

    the above

    3. The attempt at a solution

    so i get an extra ##e^{2\pi i n} ## factor, ## \theta^{m}(\tau) = \sum\limits^{\infty}_{n=0}r_{m}(\tau) q^{n} e^{2\pi i n} ##

    I think it should be obvious what to do now, but I don't know what to do next?

    Something like defining a new Fourier coefficient, ##r'{m}= r_{m} e^{2\pi i n} ## and then since the sum is to ? but that doesn't seem proper enough?

    Many thanks.
     
  2. jcsd
  3. Dec 3, 2016 #2

    stevendaryl

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    Science Advisor

    Hmm. You know that if [itex]n[/itex] is an integer, [itex]e^{i 2n\pi} = 1[/itex]?
     
  4. Dec 3, 2016 #3
    oh my ! thank you ha :)
     
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