The asymptotics of a generalised Beta function

2015/03/13 by R. B. Paris, Paris, R. B.
Mathematics · #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #Mathematical functions and polynomials #math.CA #msc:30E15 #msc:33B15 #msc:34E05 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1503.04016

11 pages, 2 figures

arxiv created 2015/03/13 · arxiv updated 2015/03/16

Abstract

We consider the generalised Beta function introduced by Chaudhry \it et al.\/ [J. Comp. Appl. Math. \bf 78 (1997) 19--32] defined by B(x,y;p)=∫₀¹ t^x-1 (1-t)^y-1 exp [(-p)/(4t(1-t))] dt, where \Re (p)>0 and the parameters x and y are arbitrary complex numbers. The asymptotic behaviour of B(x,y;p) is obtained when (i) p large, with x and y fixed, (ii) x and p large, (iii) x, y and p large and (iv) either x or y large, with p finite. Numerical results are given to illustrate the accuracy of the formulas obtained.

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