Elliptic integral

Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude:

The incomplete elliptic integral of the first kind has following addition theorem:

The incomplete elliptic integral of the second kind has following addition theorem:

The differential equation for the elliptic integral of the first kind is

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean (Carlson 2010, 19.8).