# Elliptic Integrals of the First Kind

Suppose I have $$F(\phi(x), k) = x$$ where the elliptic integral of the first kind is defined to be $$F(\phi, k) = \int_{0}^{\phi} \frac{1}{\sqrt{1-k^2\sin(\theta)}} \, d\theta$$

How could I invert this in order to make $\phi$ the subject?

• Use Jacobi elliptic functions – alexjo Nov 11 '13 at 22:23

## 2 Answers

Maple has this (in terms of the elliptic function sn):

• That last function is more often denoted the Jacobian amplitude. Note that the Jacobian amplitude is not doubly periodic (it is quasiperiodic, however), and thus not elliptic, but composing it with trigonometric functions does yield elliptic functions. – J. M. is a poor mathematician May 17 '16 at 4:12

Just to complement GEdgar's answer (+1):

This Wikipedia article gives the definition of the Jacobi elliptic function $sn(x)$ as the inverse of the incomplete elliptic integral of the first kind.

The article gives lots of other definitions of $sn(x)$ by means of either doubly periodic meromorphic functions or alternatively in terms of theta functions, but I believe you re going to be disappointed if you want an inverse function which is easily expressible in terms of elementary functions - sorry.