# Elliptic integral

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Related to Elliptic integral: elliptic integral of the second kind
 (Math.) See Integral. one of an important class of integrals, occurring in the higher mathematics; - so called because one of the integrals expresses the length of an arc of an ellipse.

Webster's Revised Unabridged Dictionary, published 1913 by G. & C. Merriam Co.
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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.
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.
where F, E, and [PI] are the first, second, and third kind of complete elliptic integral, respectively, [28, 29], in which K =
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.
7.1 Elliptic Integral Formulation for Prolate Spheroids
Integration yields [10,11] [square root of [[eta].sub.o][a.sub.s]][lambda]t = 2/[square root of [x.sub.3] - [x.sub.1]] F([[phi].sub.i], [k.sub.i]), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is elliptic integral of the first kind, [sin.sup.2] [[phi].sub.i] = [[delta].sub.2]/[[delta].sub.1] x [[[x.sub.2] - x]/[[x.sub.3] - x]], [k.sup.2.sub.i] = [[delta].sub.1]/[[delta].sub.2], [[delta].sub.1] = [x.sub.2] - [x.sub.1], [[delta].sub.2] = [x.sub.3] - [x.sub.1].
where the modulus of elliptic integral is [k.sub.0] = sin([chi][pi]/2).
The period of the closed orbit is [T.sub.k] = 4[square root of [2k.sup.2] - 1]K(k), where K(k) is the complete elliptic integral of the first kind.
The incomplete elliptic integral of the first kind F is defined as
where K(k.sub.e0]) and K(k'.sub.e0])) are the complete elliptic integral of first kind with the modulus [k.sub.e0]) and the complementary modulus [k'.sub.e0]) expressed by

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