Efficient multiple-term approximations for the generalised elliptic-type integrals
M. El-Gabali
(Received 12 December 1994; revised 27 February 1995)
Abstract
The generalised elliptic-type integral Rm(k, a, g)
ó p cos 2a - 1 (q / 2) sin 2g - 2a - 1 (q / 2) Rm(k, a, g) = ô ¾¾¾¾¾¾¾¾¾¾¾¾¾ dq
õ 0 (1 - k 2 cos q) m + 1/2
where 0 £ k < 1, Re(g) > Re(a) > 0, Re(m) > -0.5, is represented in terms of the Gauss hypergeometric function by
Kalla, Conde and Hubbell [8]. In 1987, Kalla, Lubner and Hubbell derived a simple-structured single-term approximation for this function in the
neighbourhood of k2 = 1 in some range of the parameters a, g and m. Another formula which complements the parameter range
was recently derived by the author. In this paper a novel technique is used in deriving multiple-term efficient approximations in the neighbourhood
of k2 = 1 for Rm(k, a, g) which may be considered as a generalisation to the concept of the single-term
approximations mentioned above. Two non-overlapping expressions which almost cover the entire range of parameters (k, a, g) are derived.
Closed-form solutions are obtained for single- and double-term approximations (in the neighbourhood of k2 = 1). Results show that
the proposed technique is superior to existing approximations for the same number of terms. Our formulation has potential application for a wide
class of special functions.
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Author
M. El-Gabali
Kuwait University, Department of Mathematics, P.O. Box 5969 Safat 13060 Kuwait.
Editor JAMSB(E): editor at anziamj.austms.org.au
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