Uniform Asymptotic Methods for Integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.Comment: 31 pages, 3 figure

Would You Believe It

1 Kings 17:8-16; Mark 12:41-4

American Indians and Christian Missions: Studies in Cultural Conflict

Reviewed Book: Bowden, Henry Warner. American Indians and Christian Missions: Studies in Cultural Conflict. [S.l.]: Univ of Chicago Press, 1981. Chicago History of American Religion

Thousand to one

Exodus 20:1-1

Asymptotics of a 3F2{}_3F_2 polynomial associated with the Catalan-Larcombe-French sequence

The large $n$ behaviour of the hypergeometric polynomial \FFF{-n}{\sfrac12}{\sfrac12}{\sfrac12-n}{\sfrac12-n}{-1} is considered by using integral representations of this polynomial. This ${}_3F_2$ polynomial is associated with the Catalan-Larcombe-French sequence. Several other representations are mentioned, with references to the literature, and another asymptotic method is described by using a generating function of the sequence. The results are similar to those obtained by Clark (2004) who used a binomial sum for obtaining an asymptotic expansion.Comment: 10 pages, 1 figure. Accepted for publication in {\em Analysis and Applications

Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters

We consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform equation by applying techniques used for the existing uniform expansions for gamma(a,z) and Gamma(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section

Remarks on Slater's asymptotic expansions of Kummer functions for large values of the a−a-parameter

In Slater's 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large values of the $a-$parameter. We show how similar expansions can be derived by using integral representations, and we observe discrepancies with Slater's expansions.Comment: To appear in Advances in Dynamical Systems and Applications. Proceedings of the International Conference on Differential Equations, Difference Equations and Special Functions, Patras, Greece, September 3 - 9, 2012, dedicated to the memory of Panayiotis D. Siafarikas. 13 page

Numerical and Asymptotic Aspects of Parabolic Cylinder Functions

Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.Comment: 16 pages, 1 figur