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:
and can be computed in terms of the arithmetic-geometric mean.
- The complete elliptic integral of the first kind K is defined as
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.