Sppose that a1ar and b1br are the zeros and poles

Question

Suppose that a1,...,ar and b1,...,br are the zeros and poles, respectively, in the fundamental parallelogram of an elliptic function f. Show that

a₁ + ... + a_r - b₁ - ... - b_r =nw₁ + mw₂

for some integers n and m.

[Hint: If the boundary of the parallelogram contains no zeros or poles, simply integrate zf' (z)/f(z) over that boundary, and observe that the integral of f' (z)/f(z) over a side is an integer multiple of 2πi.

If there are zeros or poles on the side of the parallelogram, translate it by a small amount to reduce the problem to the first case.]