= Duplication formula for the Weierstrass elliptic function
{title2=$\wp(2z)=\tfrac14(\wp''(z)/\wp'(z))^2-2\wp(z)$}
The <Weierstrass elliptic function> satisfies the displayed <meromorphic function> identity. To prove it, the <Riemann-Hurwitz formula> shows that its derivative has exactly three simple zeros modulo its <period lattice>, at the nonzero <two-torsion points of a complex torus>. At each such half-period $h$, evenness around $h$ gives $\wp''/\wp'=1/(z-h)+O(z-h)$, so the right side has principal part $1/(4(z-h)^2)$, matching $\wp(2z)$. At zero, $\wp=z^{-2}+O(z^2)$ and $\wp''/\wp'=-3/z+O(z^3)$ give the same principal part and zero constant term. The difference is a holomorphic <elliptic function>, hence constant by compactness, and its limit at zero is zero.
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