Elliptic Integral - Incomplete Elliptic Integral of The Third Kind

The incomplete elliptic integral of the third kind Π is

\Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac{1}{1-n\sin^2 \theta} \frac {d\theta}{\sqrt{1-(\sin\theta\sin \alpha)^2}}, or
\Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2} \frac{dt}{\sqrt{(1-m t^2)(1-t^2) }}.

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value is infinite, for any m.

A relation with the Jacobian elliptic functions is

The meridian arc length from the equator to latitude is also related to a special case of Π:

Read more about this topic:  Elliptic Integral

Famous quotes containing the words incomplete, integral and/or kind:

    “The real weakness of England lies, not in incomplete armaments or unfortified coasts, not in the poverty that creeps through sunless lanes, or the drunkenness that brawls in loathsome courts, but simply in the fact that her ideals are emotional and not intellectual.”
    —Oscar Wilde (1854–1900)

    “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)

    “And therefore, as when there is a controversy in an account, the parties must by their own accord, set up for right Reason, the Reason of some Arbitrator, or Judge, to whose sentence, they will both stand, or their controversy must either come to blows, or be undecided, for want of a right Reason constituted by Nature; so is it also in all debates of what kind soever.”
    —Thomas Hobbes (1579–1688)