Asymptotic expansion of n-dimensional Faxén-type integrals

The asymptotic expansion of n-dimensional extensions of Faxén’s integral In(z) are derived for large complex values of the variable z. The theory relies on the asymptotics of the generalised hypergeometric, orWright, function. The coefficients in the exponential expansion are obtained by means of an algorithm applicable for arbitrary n. Numerical examples are given to illustrate the accuracy of the expansions

The expansion of the confluent hypergeometric function on the positive real axis

The asymptotic expansion of the Kummer function 1F1(a; b; z) is examined as z → +∞ on the Stokes line arg z = 0. The correct form of the subdominant algebraic contribution is obtained for non-integer a. Numerical results demonstrating the accuracy of the expansion are given

The expansion of a finite number of terms of the Gauss hypergeometric function of unit argument and the Landau constants

We obtain convergent inverse factorial expansions for the sum Sn(a, b; c) of the first n ≥ 1 terms of the Gauss hypergeometric function 2F1(a, b; c; 1) of unit argument. The form of these expansions depends on the location of the parametric excess s := c−a−b in the complex s-plane. The leading behaviour as n → ∞ agrees with previous results in the literature. The case a = b = 1/2 , c = 1 corresponds to the Landau contants for which an expansion is obtained

The asymptotic expansion of a generalised Mathieu series

We obtain the asymptotic expansion of a generalised Mathieu series and its alternating variant for large complex values of the variable by means of a Mellin transform approach. Numerical examples are presented to demonstrate the accuracy of these expansions

The discrete analogue of Laplace’s method

We give a justification of the discrete analogue of Laplace’s method applied to the asymptotic estimation of sums consisting of positive terms. The case considered is the series related to the hypergeometric function pFq−1(x) (with q≥p+1) as x→+∞ discussed by Stokes [G.G. Stokes, Note on the determination of arbitrary constants which appear as multipliers of semi-convergent series, Proc. Camb. Phil. Soc. 6 (1889) 362–366]. Two examples are given in which it is shown how higher order terms in the asymptotic expansion may be derived by this procedure