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 It might be outdated or ideologically biased.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.
References in periodicals archive ?
where l, [dl.sub.M] are the contour of the meridian section and its element with the center in the point M; Q, U [member of] l are the observation point and the point with current coordinated; [[sigma].sub.m](U) is the surface density of fictitious magnetic charges; [[micro].sub.0] is the magnetic constant; K(k) is the complete elliptic integral of the first kind of module k [11];
We notice that the relations (63) and (64) can be written by using elliptic integrals. Indeed, we have E(u, m) = E(am u, m) [28], where am u = [phi] is the Jacobi amplitude and E([phi],m) = [[integral].sp.[theta].sub.0] [square root of 1 - [m.sup.2][sin.sup.2][theta]]] d[theta] is the elliptic integral of the second kind with the modulus m.
Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin, Germany, 2013.
One of the used practice for axisymmetric boundary element methods is to use polynomial approximation of the complete elliptic integrals
An analytic solution in this case was derived by Rosenthal [33] and Princen [34] in terms of incomplete elliptic integrals and is summarized in [section]7.1.
where the modulus of elliptic integral is [k.sub.0] = sin([chi][pi]/2).
In (21) and (22), the complete elliptic integrals E and E2 are defined by
In general, elliptic integrals cannot be expressed in terms of elementary functions.
where K(k.sub.e0]) and K(k'.sub.e0])) are the complete elliptic integral of first kind with the modulus [k.sub.e0]) and the complementary modulus [k'.sub.e0]) expressed by
} and the elliptic integrals K(r),K' (r) of the first kind are
where [lambda] := r/l [member of] (0,1), and F([phi],m), E([phi],m) denote the incomplete elliptic integrals of the first and second kind, respectively.