Legendre relation for Weierstrass quasi-periods (${\eta }_1{\omega }_2-{\eta }_2{\omega }_1=2\pi i$)

For an oriented lattice basis with $\mathrm{Im}({\omega }_2/{\omega }_1)>0$, write ${\eta }_j=\zeta (z+{\omega }_j)-\zeta (z)$. These differences are constant because ${\zeta }^{\prime }=-\wp$ 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.