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:
“Could slavery suggest a more complete servility than some of these journals exhibit? Is there any dust which their conduct does not lick, and make fouler still with its slime?”
—Henry David Thoreau (1817–1862)
“... 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)
“All I really want to be is boring. When people talk about me, I’d like them to say, “Carol’s basically a short Bill Bradley.” Or, “Carol’s kind of like Al Gore in a skirt.””
—Carol Moseley-Braun (b. 1947)