What is "elliptic integral"

Elliptic \El*lip"tic\, Elliptical \El*lip"tic*al\, a. [Gr. ?: cf. F. elliptique. See Ellipsis.]

1. Of or pertaining to an ellipse; having the form of an ellipse; oblong, with rounded ends.

   The planets move in elliptic orbits.
   --Cheyne.

   The billiard sharp who any one catches, His doom's extremely hard He's made to dwell In a dungeon cell On a spot that's always barred. And there he plays extravagant matches In fitless finger-stalls On a cloth untrue With a twisted cue And elliptical billiard balls!
   --Gilbert and Sullivan (The Mikado: The More Humane Mikado Song)

2. Having a part omitted; as, an elliptical phrase.

3. leaving out information essential to comprehension; so concise as to be difficult to understand; obscure or ambiguous; -- of speech or writing; as, an elliptical comment.

   Elliptic chuck. See under Chuck.

   Elliptic compasses, an instrument arranged for drawing ellipses.

   Elliptic function. (Math.) See Function.

   Elliptic integral. (Math.) See Integral.

   Elliptic polarization. See under Polarization.

Integral \In"te*gral\, n.

1. A whole; an entire thing; a whole number; an individual.

2. (Math.) An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.

   Elliptic integral, one of an important class of integrals, occurring in the higher mathematics; -- so called because one of the integrals expresses the length of an arc of an ellipse.

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form

$f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt,$

where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant.

In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of . However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.