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.

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.
e0])) are the complete elliptic integral of first kind with the modulus [k.
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.
First, its arc length should be given by an elliptic integral, say [integral] dx/ [square root of p(x)] for a certain polynomial p.
The following formulas for derivation of complete elliptic integrals will be used:
Elliptic integrals have many applications, for example in mathematics and physics:
Consequently, the ratio of the complete elliptic integrals of first kind in Equations 29 and 30 evaluated with [P.
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:
Besides the Legendre form , the elliptic integrals may also be expressed in Carlson symmetric form.
There are a number of specialized integrators for handling elliptic integrals, integrals of algebraic functions, and integrals involving special functions, etc.
we only have to compute two complete elliptic integrals by an arithmetic geometric mean process.