The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse :
where a is the semi-major axis, and e is the eccentricity.
E may be defined as
or more compactly in terms of the incomplete integral of the second kind as
It can be expressed as a power series
which is equivalent to
In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
The complete elliptic integral of the second kind can be most efficiently computed in terms of the arithmetic-geometric mean and its modification.
Read more about this topic: Elliptic Integral
Famous quotes containing the words complete, integral and/or kind:
“It is easier to live through someone else than to become complete yourself.”
—Betty Friedan (b. 1921)
“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)
“The man who, from the beginning of his life, has been bathed at length in the soft atmosphere of a woman, in the smell of her hands, of her bosom, of her knees, of her hair, of her supple and floating clothes, ... has contracted from this contact a tender skin and a distinct accent, a kind of androgyny without which the harshest and most masculine genius remains, as far as perfection in art is concerned, an incomplete being.”
—Charles Baudelaire (1821–1867)