Definition:Elliptic Function

From ProofWiki
Jump to navigation Jump to search


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.


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).