Elliptic integral

An Elliptic integral is an integral of the type

Where R is a rational function in two variables and P ( x) is a polynomial of the third or fourth degree without multiple zero. The integral is elliptical, because of integral molding to occur in the calculation of the extent of the surface of ellipses and ellipsoids.

Elliptic integrals can not generally be represented by elementary functions, but they can be converted by transformations into a sum of elementary functions and integrals of the form described below. These integrals are called elliptic integrals of the first, second and third kind

I. Type: II Type: III. Type:

It is a result of the substitution of these integrals are brought to the Legendre form:

I. Type: II Type: III. Type:

The integrals with integral lower limit 0 is called incomplete elliptic integrals. If, in addition, the upper limit of the integral, it is called in the case of type I and II of the complete elliptic integrals

The values ​​of these integrals are tabulated.

Inverse functions or algebraic functions of inverse functions of elliptic integrals are called elliptic functions. They are related to the trigonometric functions in a certain way.