# Definition:Elliptic Function

## Definition

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

Consider the inverse of $\map y x$:

- $x = \map \phi y$

Then $\phi$ is an **elliptic function**.

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## Examples

### First Kind

Consider the incomplete elliptic integral of the first kind:

- $u = \ds \int_0^x \dfrac {\d t} {\sqrt {\paren {1 - t^2} \paren {1 - k^2 t^2} } }$

Then we have the following **elliptic functions**:

\(\ds x\) | \(=\) | \(\ds \sn u\) | ||||||||||||

\(\ds \sqrt {1 - x^2}\) | \(=\) | \(\ds \cn u\) | ||||||||||||

\(\ds \sqrt {1 - k^2 x^2}\) | \(=\) | \(\ds \dn u\) |

## Also see

- Results about
**elliptic functions**can be found**here**.

## Historical Note

**Elliptic functions** were first explored by Niels Henrik Abelâ€Ž in $1827$, after his discovery of them as the inverses 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.*

Charles Hermite used **elliptic functions** in $1858$ in his solution of the general quintic equation.

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 ($\text {1815}$ – $\text {1897}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**elliptic functions** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**elliptic functions**