Duplication Formula for Weierstrass's Elliptic Function

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Theorem

Let $\omega_1$, $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.

Let $z$ be a complex number where $z \notin \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$.

Then:

$\map \wp {2 z; \omega_1, \omega_2} = \dfrac 1 4 \paren {\dfrac {\map {\wp} {z; \omega_1, \omega_2} } {\map {\wp'} {z; \omega_1, \omega_2} } }^2 - 2 \map \wp {z; \omega_1, \omega_2}$

where:

$\wp$ is Weierstrass's elliptic function

$\wp'$ and $\wp$ denote its first and second derivative with respect to $z$.

Proof

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Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $20.311$: The duplication formula for $\wp \left({z}\right)$