= Weierstrass half-period translation formula
{c}
{title2=$\wp(z+h_i)=e_i+\frac{(e_i-e_j)(e_i-e_l)}{\wp(z)-e_i}$}
Let $h_i$ represent a nonzero <two-torsion point of a complex torus>, and write $e_i=\wp(h_i)$. The product $(\wp(z+h_i)-e_i)(\wp(z)-e_i)$ is an <elliptic function> whose apparent <poles> cancel against double <zeros>. It is therefore constant by <Liouville theorem>. Its value at $z=0$ is $\wp''(h_i)/2$. The <Weierstrass elliptic differential equation> gives $\wp''(h_i)/2=3e_i^2-g_2/4=(e_i-e_j)(e_i-e_l)$, proving the formula.
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