The complete elliptic integral of the first kind K may be defined as

or

and can be computed in terms of the arithmetic-geometric mean. In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ...

It is a special case of the incomplete elliptic integral of the first kind: In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

It can also be calculated as

In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...

The complete elliptic integral of the first kind is sometimes called the quarter period. In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ...