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:
Read more about this topic: Elliptic Integral
Famous quotes containing the words complete, integral and/or kind:
“No man, said Birkin, cuts another man’s throat unless he wants to cut it, and unless the other man wants it cutting. This is a complete truth. It takes two people to make a murder: a murderer and a murderee.... And a man who is murderable is a man who has in a profound if hidden lust desires to be murdered.”
—D.H. (David Herbert)
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (1533–1592)
“Each person calls barbarism whatever is not his or her own practice.... We may call Cannibals barbarians, in respect to the rules of reason, but not in respect to ourselves, who surpass them in every kind of barbarity.”
—Michel de Montaigne (1533–1592)