= Legendre relation for Weierstrass quasi-periods
{c}
{title2=$\eta_1\omega_2-\eta_2\omega_1=2\pi i$}
For an oriented lattice basis with $\operatorname{Im}(\omega_2/\omega_1)>0$, write $\eta_j=\zeta(z+\omega_j)-\zeta(z)$. These differences are constant because $\zeta'=-\wp$ is periodic. Integrating zeta around a fundamental cell gives the displayed relation by its one simple pole of residue one. Oddness and the logarithmic derivative then give $\sigma(z+\omega_j)=-e^{\eta_j(z+\omega_j/2)}\sigma(z)$. This fixes both the sign and the exponent convention in sigma-product constructions.
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