Narrow Results

I trace the main steps of the first fifty-five years of my career as an applied mathematician, pausing from time to time to describe problems that arose in asymptotics and numerical analysis and had far-reaching effects on this career.

Lecture delivered at Asymptotics and Applied Analysis, Conference in Honor of Frank W. J. Olver's 75th Birthday, January 10–14, 2000, San Diego State University, San Diego, California.

Editors' Note: Frank W. J. Olver died on April 23, 2013. The following text was typed by his son, Peter J. Olver, from handwritten notes found among his papers. At times the writing is unpolished, including incomplete sentences, but the editors have decided to leave it essentially the way it was written. However, for clarity, some abbreviations have been written out in full. A couple of handwritten words could not be deciphered, and a guess for what was intended is enclosed in brackets: […]. Endnotes have been made into footnotes within the body of the article. References were mostly not included in the handwritten text, but rather listed in order at the end. Citations to references have been included at the appropriate point in the text.

The aim of this paper is to derive new representations for the Hankel and Bessel functions, exploiting the reformulation of the method of steepest descents by Berry and Howls [Hyperasymptotics for integrals with saddles, Proc. R. Soc. Lond. A434 (1991) 657–675]. Using these representations, we obtain a number of properties of the large-order asymptotic expansions of the Hankel and Bessel functions due to Debye, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

In this paper, we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Hyperasymptotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A439 (1992) 373–396]. Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

In this chapter, six of seven mock theta-functions of the third order of Ramanujan are expressed in terms of Appell–Lerch series and generalized Mordell integrals.

The development of asymptotic expansions of Stieltjes transforms of exponentially decaying functions has been well established. In this paper, we are concerned with the more difficult case in which the functions decay only algebraically at infinity. By using a Gevrey-type condition, we obtain an exponentially improved asymptotic expansion, and give three representation theorems to show that the Stieltjes transform of algebraically decaying functions can be written as the difference of two integral transforms with exponentially decaying kernels, thus making the asymptotic theory developed for integral transforms with exponentially decaying kernels relevant to Stieltjes transforms of algebraically decaying functions, including the smoothing of the Stokes phenomenon.