# Definition:Elliptic Function

## Definition

Let $\displaystyle y \left({x}\right) = \int_0^x \dfrac {\d t} {\sqrt {P \left({t}\right)} }$ be an elliptic integral, where $P \left({t}\right)$ is a polynomial of degree $3$ or $4$.

Consider the inverse of $y \left({x}\right)$:

$x = \phi \left({y}\right)$

Then $\phi$ is an elliptic function.

## Historical Note

Elliptic functions were first explored by Niels Henrik Abel‎ in $1827$, after his discovery of them as the inverse of elliptic integrals.

Carl Gustav Jacob Jacobi then continued the work in $\text {1828}$ – $\text {1829}$.

However, it turned out that Carl Friedrich Gauss had actually got there first, but had never got round to publishing his work.

Jacobi noticed a passage in Gauss's Disquisitiones Arithmeticae (Article $335$) in which it was clear that Gauss' had already arrived at the same results that Jacobi had done, but some $30$ years before.

As Jacobi wrote to his brother:

Mathematics would be in a very different position if practical astronomy had not diverted this colossal genius from his glorious career.

Joseph Liouville based his own theory of elliptic functions on his Liouville's Theorem (Complex Analysis).