# Regarding property of odd elliptic functions

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 .

Let $$L \subseteq \mathbb{C}$$ be a lattice and let $$f$$ be an odd elliptic function with respect to $$L$$. Recall that this means that $$f(z+\omega) = f(z)$$ for all $$z \in \mathbb{C}, \omega \in L$$ such that $$f$$ does not have a pole at $$z$$.
Given any $$\omega \in L$$, there are precisely two possibilities:
(i) $$\omega/2$$ is a pole of $$f$$.
(ii) $$\omega/2$$ is not a pole of $$f$$. In this case, we may conclude that $$f(\omega/2) = f(\omega/2 - \omega) = f(- \omega/2) = - f(\omega/2),$$ where we used at the first '$$=$$' that ($$\omega \in L$$ and hence) $$-\omega \in L$$, and at the third '$$=$$' that $$f$$ is odd. This now implies that $$2 f(\omega/2) = 0$$ and hence that $$f(\omega/2) = 0$$.