I am working on the following problem that was previously posted regarding the integral of an elliptic function. I understand the entire solution that was posted; however, there is a small piece of the puzzle that I do not understand.
We have shown that $S(0)-S(\infty)=\int_{\partial\mathcal{F}}z\frac{f'(z)}{f(z)}dz$, and we wish to show that this is a period of $f$ that is we wish to show that it is an element of the lattice. I note that
$$\int_{\partial\mathcal{F}}z\frac{f'(z)}{f(z)}dz=\int_a^{a+\omega_1}z\frac{f'(z)}{f(z)}dz+\int_{a+\omega_1}^{a+\omega_1+\omega_2}z\frac{f'(z)}{f(z)}dz+\int_{a+\omega_1+\omega_2}^{a+\omega_2}z\frac{f'(z)}{f(z)}dz+\int_{a+\omega_2}^az\frac{f'(z)}{f(z)}dz$$
Now using the periodicity of $f$ we have that $$ \int_a^{a+\omega_1}(z+\omega_2)\frac{f'(z)}{f(z)}dz=\int_{a+\omega_2}^{a+\omega_1+\omega_2}z\frac{f'(z)}{f(z)}dz $$ and $$ \int_a^{a+\omega_2}(z+\omega_1)\frac{f'(z)}{f(z)}dz=\int_{a+\omega_2}^{a+\omega_1+\omega_2}z\frac{f'(z)}{f(z)}dz $$ Thus, when we simplify make this substitution we should arrive at $$ S(0)-S(\infty)=-\omega_1\int_a^{a+\omega_1}z\frac{f'(z)}{f(z)}dz-\omega_2\int_a^{a+\omega_2}\frac{f'(z)}{f(z)}dz $$ The place I am getting stuck is I want to show that both of these integrals are integers as that would let us conclude the result, and I think it might be some application of the residue theorem, but these are not closed curves, so I am stuck at this last step, any help would be appreciated.