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:

    “Each of us is incomplete compared to someone else, an animal’s incomplete compared to a person ... and a person compared to God, who is complete only to be imaginary.”
    —Georges Bataille (1897–1962)

    “... 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)

    “I do not know what the spirit of a philosopher could more wish to be than a good dancer. For the dance is his ideal, also his fine art, finally also the only kind of piety he knows, his “divine service.””
    —Friedrich Nietzsche (1844–1900)