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Elliptic Functions, same principal parts, finding additive C

  1. Apr 13, 2017 #1
    1. The problem statement, all variables and given/known data

    See attached.

    ellipcpc3.png

    The solution of part e) is ##C=4\psi(a)##

    I am looking at part e, the answer to part d being that the principal parts around the poles ##z=0## and ##z=-a## are the same.

    2. Relevant equations


    3. The attempt at a solution

    Since we already know the negative powers of ##z## have the same expansions, and ##C## corresponds to the ##z^0## term, ##f_a(z)^2## about ##z=0## gives ##\frac{4}{z^2}+4\psi(a)z^2+4\psi(a)## and so the relevant term is ##4\psi(a)##.

    Looking at the expansion of ##f_a(z)^2## about ##z=-a## there is no ##z^0## term so I conclude ##C=4\psi(a)##.

    QUESTION
    - This doesnt really seem like a proper approach, i.e to break it down to considering the expansions of ##f_a(z)## about ##z=0 ## and ##z=-a## separately, whereas I am considering the RHS as a function over the entire complex plane . ( if I wasn't considering the RHS or LHS over the entire complex plane then the theorem to give the additive constant ##C## does not work, so I don't really see how you can break it down on either the LHS or RHS to consider only an expansion about a single pole ? )

    Thanks in advance.
     
  2. jcsd
  3. Apr 18, 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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