# Elliptic Functions, same principal parts, finding additive C

1. Apr 13, 2017

### binbagsss

1. The problem statement, all variables and given/known data

See attached.

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 ? )