Elliptic Integrals are said to be 'complete' when the amplitude φ=π/2 and therefore x=1. The complete elliptic integral of the first kind K may thus be defined as
or more compactly in terms of the incomplete integral of the first kind as
It can be expressed as a power series
where Pn is the Legendre polynomial, which is equivalent to
where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
The complete elliptic integral of the first kind is sometimes called the quarter period. It can most efficiently be computed in terms of the arithmetic-geometric mean:
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