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 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.