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 see advertisements for active young men, as if activity were the whole of a young man’s capital. Yet I have been surprised when one has with confidence proposed to me, a grown man, to embark in some enterprise of his, as if I had absolutely nothing to do, my life having been a complete failure hitherto. What a doubtful compliment this to pay me!”
—Henry David Thoreau (1817–1862)
“Self-centeredness is a natural outgrowth of one of the toddler’s major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, what’s his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of him—an integral piece—is being torn from him.”
—Lawrence Balter (20th century)
“There are acacias, a graceful species amusingly devitalized by sentimentality, this kind drooping its leaves with the grace of a young widow bowed in controllable grief, this one obscuring them with a smooth silver as of placid tears. They please, like the minor French novelists of the eighteenth century, by suggesting a universe in which nothing cuts deep.”
—Rebecca West (1892–1983)