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.
 | This article is complete as far as it goes, but it could do with expansion. In particular: Serious lots of filling out to be done here. I will do what I can in due course with the sources I have. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 Abel‎ 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