# 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?