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Mellin transform-- complete Riemann function

  1. Dec 14, 2016 #1
    1. The problem statement, all variables and given/known data

    Show that ## \int\limits^{\infty}_0 \sum\limits^{\infty}_{n=1}e^{-\pi n^{2} t} t^{s-1} dt = \sum\limits^{\infty}_{n=1}(\pi n^{2} )^{-s} \int\limits^{\infty}_{0} e^{-t} t^{s-1} dt ##

    2. Relevant equations

    ##\Phi (t)=\theta(it/2)=1+2\sum\limits^{\infty}_{n=1}e^{-\pi n^2 t}. ##

    With the transformation formula : ##\Phi(1/t)=t^{1/2}\Phi(t) ##

    3. The attempt at a solution

    I am unsure really where to start, a clue would be greatly appreciated.

    The only thing I can really think of is transforming ##t## to get ##t^{s}## rather than ##t^{1/2}## on the RHS of (1), but I am a) unsure how to do this b) unsure whether it is the right thing to do

    The other thing would be to expand out the exponential, writing it as two expressions multiplied together corresponding to the ##-t## in the exponent power and the ## \pi n^{2} ## seperately, collecting the sum to infinity to get ##e^{-t}## and leaving the bit corresponding to "##e^{\pi n^{2}}" ## as a summation. However the ##-s## means the choice of my variable of this summation would need to be ##s##, which doesn't really make sense?

    I'm quite stuck.

    Any help greatly appreciated,

    Many thanks.
     
    Last edited: Dec 14, 2016
  2. jcsd
  3. Dec 14, 2016 #2
    Do a change of variables of the form ##u=\pi n^2 t##.
     
  4. Dec 14, 2016 #3
    cheers !
     
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