While self studying analytic number theory from Tom M Apostol modular functions and Dirichlet series in number theory I am unable to think about an argument which Apostol doesn't proves but uses it in Theorem 1.14 of chapter - elliptic functions.
This same problem was also asked in lecture notes on elliptic functions which I was studying.
Half periods of odd elliptic functions are either zeroes or poles .
I have doubt in only the part when $\omega$ /2 is proved to not to be zero. How to prove such half period must be pole .
Can someone please help on how to prove it.