# 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 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 ?
Q] is the unit normal to the contour l in the point Q [member of] l; E(k), k is the complete elliptic integral of the second kind of the module k and the additional module of complete elliptic integrals [9]; k' =[square root of (1 - [k.
Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping.
elliptic integrals of first kind) are available in most mathematical software packages.
For background material on elliptic integrals the reader can consult McKean and Moll [6, chapter 2].
3] are the following combinations of elliptic integrals K, E and their argument [beta]:
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).
The generic ratio K(q)/K(q') is the ratio of the complete elliptic integrals of first kind, [15] evaluated by W.
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
where L is the incomplete elliptic integral of the first kind, k is its modulus, and [psi] is its amplitude.
b]'), where K() is the complete elliptic integral of the first kind.
For the example of the third symmetric elliptic integral, considered as a motivation for this section, we have (we only show the dominant term of the expansion) [18]:

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