# elliptic integral of the first kind

## elliptic integral of the first kind

[ə¦lip·tik ¦int·ə·grəl əvthə ¦fərst ‚kīnd]
(mathematics)
Any elliptic integral which is finite for all values of the limits of integration and which approaches a finite limit when one of the limits of integration approaches infinity.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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where l, [dl.sub.M] are the contour of the meridian section and its element with the center in the point M; Q, U [member of] l are the observation point and the point with current coordinated; [[sigma].sub.m](U) is the surface density of fictitious magnetic charges; [[micro].sub.0] is the magnetic constant; K(k) is the complete elliptic integral of the first kind of module k [11];
where K(m) is the complete elliptic integral of the first kind. Note that the wavelength L given by (34) is two times greater than that for a single [cn.sup.2] periodic solution [21].
Chu, "A monotonicity property involving the generalized elliptic integral of the first kind," Mathematical Inequalities & Applications, vol.
In this paper, we investigate the square lattice using an approach for the calculation of Green's function that is based on the evaluation of a complete elliptic integral of the first kind. Unlike the previous approach, we obtain results that are applicable over the entire absorption band.
An axisymmetric fundamental solution [u.sup.*.sub.ax] is defined in terms of the complete elliptic integral of the first kind K(m), see e.g.
The incomplete elliptic integral of the first kind F is defined as
where [k'.sup.2] = 1 - [k.sup.2] is called as complementary modulus, and K(k) is the complete elliptic integral of the first kind:
where L is the incomplete elliptic integral of the first kind, k is its modulus, and [psi] is its amplitude.
A series expansion of an elliptic or hyperelliptic integral in elementary symmetric functions is given, illustrated with numerical coefficients for terms through degree seven for the symmetric elliptic integral of the first kind. Its usefulness for elliptic integrals, in particular, is important.
[q.sub.0], [q.sub.2], and [q.sub.4] are defined in (19) and K(m) is the complete elliptic integral of the first kind defined in (16).
Complete elliptic integral of the first kind is defined as Elliptic Integrals are said to be complete when the amplitude

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