Differential Equation satisfied by Weierstrass's Elliptic Function

Theorem

The differential equation:

$\paren {\dfrac {\d f} {\d z} }^2 = 4 f^3 - g_2 f - g_3$

where:

$\ds g_2 = 60 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^4}$

and:

$\ds g_3 = 140 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^6}$

has the general solution:

$\map f z = \map \wp {z + C; \omega_1, \omega_2}$

where:

$\wp$ is Weierstrass's elliptic function

$C$ is an arbitrary constant

$\omega_1$, $\omega_2$ are constants independent of $z$.

This article, or a section of it, needs explaining. In particular: What domain are all the variables and constants in? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $20.22$: The differential equation satisfied by $\map \wp z$