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:
“I want relations which are not purely personal, based on purely personal qualities; but relations based upon some unanimous accord in truth or belief, and a harmony of purpose, rather than of personality. I am weary of personality.... Let us be easy and impersonal, not forever fingering over our own souls, and the souls of our acquaintances, but trying to create a new life, a new common life, a new complete tree of life from the roots that are within us.”
—D.H. (David Herbert)
“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)
“Daddy?
That’s another kind of prison.
It’s not the prince at all,
but my father
drunkenly bent over my bed,
circling the abyss like a shark....”
—Anne Sexton (1928–1974)