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 It might be outdated or ideologically biased.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 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 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.

References in periodicals archive ?
where E([THETA] | m) is the elliptic integral of the second kind, am(x | m) is the Jacobi elliptic function amplitude, and E(m) is the complete elliptic integral of the second kind.
A calculation of [partial derivative][u.sup.*.sub.ax]/[partial derivative]n, with the use of the relation for K'(m) in terms of the complete elliptic integral of the second kind E(m), gives
The complete elliptic integral of the first kind is sometimes called the quarter period.
In the case where the spectrum is real, we can simply compute the upper and lower bounds of the spectrum by the Arnoldi (or, if F = [F.sup.T], the Lanczos) process and enter the Wachspress computation with these values for a and b, and set a = 0, i.e., we only have to compute two complete elliptic integrals by an arithmetic geometric mean process.
Algorithms for Complete Elliptic Integrals of the Third Kind
As y = [alpha] = [[chi square].sub.0]), then P(2 arccot 0,m) = P([pi],m) = 2K(m), where K(m) is the complete elliptic integral of the first kind defined as [20]
Complete elliptic integral of the first kind is defined as Elliptic Integrals are said to be complete when the amplitude
Among the topics are complete elliptic integrals, the Riemann zeta function, some automatic proofs, the error function, hyper-geometric functions, Bessel-K functions, polylogarithm functions, evaluation by series, the exponential integral, confluent hyper-geometric and Whittaker functions, and the evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets.
Here, K, K' are the complete elliptic integrals of the first kind:
As another example, let us apply Mathcad to a case of coplanar structure, which demonstrates this software's capability to treat hyperbolic trigonometric such as Tanh() and special functions (such as ratios of complete elliptic integrals).
where E(w) and K(w) are respectively complete elliptic integrals of the first and second kind, defined according to the convention, (7)

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