Elliptic integral

Elliptic integral

In integral calculus, an elliptic integral is any function f which can be expressed in the form

where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.

Particular examples include:

The complete elliptic integral of the first kind K is defined as

and can be computed in terms of the arithmetic-geometric mean.

It can also be calculated as

The complete elliptic integral of the second kind E is defined as

The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as

Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular; we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.

The origin of the name

Historically properties of these integrals were studied in connection with the problem of the arc length of an ellipse, by Fagnano and Leonhard Euler.