Elliptic Integral

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elliptic integral

[ə′lip·tik ′int·ə·grəl]
(mathematics)
An integral over x whose integrand is a rational function of x and the square root of p (x), where p (x) is a third- or fourth-degree polynomial without multiple roots.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Elliptic Integral

 

any integral of the type

R (x, y) dx

where R(x, y) is a rational function of x and Elliptic Integral being a third-or fourth-degree polynomial without multiple roots.

The integral

is called an incomplete elliptic integral of the first kind, and the integral

is called an incomplete elliptic integral of the second kind. Here, k is the modulus of the elliptic integral, 0 < k < 1 (x = sin φ, t = sin α). The integrals on the left-hand side of equations (1) and (2) are known as Jacobi’s normal forms, and the integrals on the right-hand side are known as Legendre’s normal forms. When x = 1 or φ = π/2, the elliptic integrals are said to be complete and are designated by

and

respectively.

Elliptic integrals owe their name to their appearance in the problem of calculating the length of an arc of an ellipse ua sin α v = b cos α (a > b). The length of an arc of the ellipse is expressed by the formula

where Elliptic Integral is the eccentricity of the ellipse. The length of one-fourth of the circumference of an ellipse is equal to E(k). The inverse functions of elliptic integrals are called elliptic functions.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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Complete elliptic integral of the first kind is defined as Elliptic Integrals are said to be complete when the amplitude
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