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:

    “I dont think there is anything on earth more wonderful than those wistful incomplete friendships one makes now and then in an hour’s talk. You never see the people again, but the lingering sense of their presence in the world is like the glow of an unseen city at night—makes you feel the teemingness of it all.”
    —John Dos Passos (1896–1970)

    “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 kind uncles and aunts of the race are more esteemed than its true spiritual fathers and mothers.”
    —Henry David Thoreau (1817–1862)