An
elliptic integral is any
function f which can be expressed in the form
- <math> f(x) = \int_{c}^{x} R(t,P(t))\ dt </math>
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
- <math> K(x) = \int_{0}^{1} \frac{1}{ \sqrt{(1-t^2)(1-x^2 t^2)} }\ dt </math>
- and can be computed in terms of the arithmetic-geometric mean.
- The complete elliptic integral of the second kind E is defined as
- <math> E(x) = \int_{0}^{1} \frac{ \sqrt{1-x^2 t^2} }{ \sqrt{1-t^2} }\ dt </math>