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Elliptic functions: Weierstrass psi function limit

  1. Jul 19, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that
    ##\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2})) =1##

    where ##\psi(z)=\frac{1}{z^2}+\sum\limits_{w \in \Omega}' \frac{1}{(z-w)^2}-\frac{1}{w^2}##

    where ##\Omega## are the periods of ##\psi(z)##

    2. Relevant equations

    3. The attempt at a solution

    ##\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2}))= 1 + \sum\limits_{w\in \Omega}'\frac{z^2}{(z-w)^2}-\frac{z^2}{w^2}##

    The last time clearly vanishes.

    For the second term I can write this as ##\frac{1}{(1-\frac{w}{z})^2} \to 1 ## as ##z \to 0 ##

    So I get

    ##\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2})) =1+ \sum_{w \in \Omega}' 1##

    This is probably a stupid question but I don't really understand the summation second term here.

    ##\sum_{n=1}^{n=n} 1 = n## right?

    So there's an infintie number of ## w \in \Omega ## so obviously I dont want to do this, are you in affect looking at the limit '##mod \Omega ##', so where the first term corresponds to ##w=0 ## ?

    Many thanks
     
  2. jcsd
  3. Jul 24, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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