Elliptic Integral


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Related to Elliptic Integral: elliptic integral of the second kind

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.

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.

References in periodicals archive ?
Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping.
3] are the following combinations of elliptic integrals K, E and their argument [beta]:
Here, the functions F([phi],k) and E([phi],k) are the incomplete elliptic integrals of the first and second kind, respectively (Weisstein 2007a, 2007b).
Complications formerly encountered in numerical computation of Legendre's complete elliptic integral of the third kind were avoided by defining and tabulating Heuman's Lambda function (for circular cases) and a modification of Jacobi's Zeta function (for hyperbolic cases).
From series development of the elliptic integrals for p [much less than] 1 and p[prime] [much less than] 1, p and c can be approximated by the following equations:
Complete elliptic integral of the second kind is defined as The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse C
In the case of real spectra, [phi] = [pi] / 2 and L[[pi] / 2, k] is a complete elliptic integral of the form
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.
Subjects include fractional integration and fractional differentiation for d- dimensional Jacobi expansions, Sutherland-type trigonometric models, a generating function for N-soliton solutions of the Kadomtsev- Petviashvili II equation, asymptotics of the second Painleve equation, evaluation of certain Mellin transformations in terms of the trigamma and polygamma functions, conformal maps to generalized quadrature domains, approximations for zeros of Hermite functions, qualities and bounds for elliptic integrals, P-symbols in Huen identities, iterative method for numerical integration of rational functions, a Taylor expansion theorem for any elliptic extension of the Askey-Wilson operator, and Ramanujan's symmetric theta functions in his lost notebook.
S (Approximations of Special Functions) has new routines for polygamma functions, zeros of Bessel functions, jacobian functions, elliptic integrals and associations Legendre functions.

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