# Definition:Elliptic Function

## Contents

## 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**.

## Also see

## 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 $1828$ – $1829$.

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).

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($1815$ – $1897$)