# Duplication Formula for Weierstrass's Elliptic Function

## 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 $z \notin \left\{ {2 m \omega_1 + 2 n \omega_2: \left({n, m}\right) \in \Z^2}\right\}$.

Then:

- $\displaystyle \wp \left({2 z; \omega_1, \omega_2}\right) = \frac 1 4 \left({ \frac {\wp'' \left({z; \omega_1, \omega_2}\right)} {\wp' \left({z; \omega_1, \omega_2}\right)} }\right)^2 - 2 \wp \left({z; \omega_1, \omega_2}\right)$

where:

- $\wp$ is Weierstrass's elliptic function
- $\wp'$ and $\wp''$ denote its first and second derivative with respect to $z$.

## Proof

## Sources

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