= Zero-pole sum of an elliptic function
{title2=$\sum_j a_j-\sum_j b_j\in\Lambda$}
For a nonzero <elliptic function> with <period lattice> $\Lambda$, its <zeros of a function> $a_j$ and <poles> $b_j$, repeated with their <multiplicities>, have equal total number and satisfy $\sum_j a_j-\sum_j b_j\in\Lambda$. Choose a fundamental parallelogram avoiding all zeros and poles on its boundary. The <argument principle> applied to $f'/f$ gives equality of the numbers. Applying the <residue theorem> to $zf'/f$ gives their difference of sums. Pairing opposite edges, the extra factors are the two lattice generators multiplied by integrals of $f'/f$ along an edge. These integrals are integer multiples of $2\pi i$, because the endpoints have the same nonzero function value. Thus the difference of sums is a lattice element. This links zeros of a pulled-back line to the <chord-and-tangent group law>.
Back to article page
