Homework Help: Elliptic functions proof f(z)-c has N zeros, N the order

1. Jun 28, 2017

binbagsss

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

proof of theorem

2. Relevant equations

3. The attempt at a solution

Hi,
I have a couple of questions on the attached proof and theorem

1) On the last line, how is it we go from the order of the zeros = the number of zeros, or is it's meaning the number of zeros counted with multiplicity. - Am I correct in thinking that:

order of pole/zero = number of poles/zeros * multiplicity at each point ?

2) The zero expansion considered is fine since we are considering zeros of $f(z)-c$. But for expansion about a pole of $f(z)$, near that pole, how is it we can say that that same expansion holds for $f(z)-c$? Is this not making some assumptions on $c$ being small?

Many thank in advance.

2. Jul 3, 2017

PF_Help_Bot

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.