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Elliptic integral

An elliptic integral is any function f which can be expressed in the form

$f(x) = \int_{c}^{x} R(t,P(t))\ dt$

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
$K(x) = \int_{0}^{1} \frac{1}{ \sqrt{(1-t^2)(1-x^2 t^2)} }\ dt$
and can be computed in terms of the arithmetic-geometric mean.

• The complete elliptic integral of the second kind E is defined as
$E(x) = \int_{0}^{1} \frac{ \sqrt{1-x^2 t^2} }{ \sqrt{1-t^2} }\ dt$