Elliptic Integral

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Related to Elliptic integrals: elliptic integral of the second kind, elliptic integral of the third kind, elliptic integral of the first 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 ?
In general, elliptic integrals cannot be expressed in terms of elementary functions.
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.
The following formulas for derivation of complete elliptic integrals will be used:
Handbook of elliptic integrals for engineers and physicists.
Elliptic integrals have many applications, for example in mathematics and physics:
where, k(p[prime]) is the complete elliptic integral of first kind, p[prime] = [square root of 1 - [p.
Complete elliptic integral of the first kind is defined as Elliptic Integrals are said to be complete when the amplitude
The new as well as the Wachspress parameter algorithms require the computation of certain elliptic integrals presented in (2.
where E(w) and K(w) are respectively complete elliptic integrals of the first and second kind, defined according to the convention, (7)
2] Hancock Harris, Elliptic integrals, John Wiley&sons, Inc.
Among these are the Bessel functions, hypergeometric functions, elliptic integrals, probability distributions, and orthogonal polynomials.