Elliptic integrals and the arithmetic-geometric mean

In this entry we introduce the Arithmetic Geometric mean and show it´s connection with the complete elliptic integral of the first kind.

$\begin{align*} K(k)&=\frac{\pi}{2\operatorname{M}\left(1,k^\prime \right)} \end{align*}$

This relation is one way to establish the connection between the Jacobi theta function and elliptic integral.

$\begin{align*} \operatorname{M}\left(1,k^\prime \right)&=\frac{1}{\vartheta_3^2(q)} \end{align*}$

and

$\begin{align*} \vartheta_3(q)&=\sqrt{\frac{2K(k)}{\pi}} \end{align*}$

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Important to note that in this entry we wont get into a detailed study of elliptic integral.

Commentaries and feedbacks are welcome.

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