\documentclass[11pt,leqno]{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} %\renewcommand{\qedsymbol}{\rule{1.3mm}{2.8mm}} %\usepackage{epsf} \topmargin 0.00in \oddsidemargin 0.00in \evensidemargin 0.00in \marginparwidth 0.00in \marginparsep 0.00in \textwidth 6.27in \textheight 8.69in \linespread{1.5} \pagestyle{plain} \begin{document} \title{EXPONENTIAL ASYMPTOTICS} \bigskip \author{R. Wong} \date{Department of Mathematics\\ City University of Hong Kong\\ Tat Chee Avenue\\ Kowloon, Hong Kong} \maketitle \bigskip \bigskip \vspace{30mm} \bigskip \centerline{\large NATO/ASI \qquad Special Functions 2000} \newpage Stokes phenomenon occurs typically in compound asymptotic expansions; it concerns the abrupt change in the coefficients of these expansions, when the variable crosses certain lines in the complex plane. To illustrate this phenomenon, we consider the Airy integral \begin{equation} \text{Ai}(z)=\frac{1}{2\pi i}\underset{L}{\int}\text{exp}(\tfrac{1}{3} t^3-zt)dt, \end{equation} where $L$ is any contour which begins at infinity in the sector $-\frac{1}{2}\pi < \text{art}\; t<-\frac{1}{6}\pi$ and ends at infinity in the sector $\frac{1}{6}\pi < \text{arg}\:t<\frac{1}{2}\pi$. From (1), it can be easily verified that $\text{Ai}(z)$ has the Maclaurin expansion $$\text{Ai}(z)=3^{-2/3}\sum^{\infty}_{n=0}\frac{z^{3n}}{3^{2n}n!\Gamma(n+\frac{2}{3})}\:-3^{-4/3}\sum^{\infty}_{n=0}\:\frac{z^{3n+1}}{3^{2n}n!\Gamma(n+\frac{4}{3})}.$$ Thus, $\text{Ai}(z)$ is an entire function. If we denote by $u_+(z)$ and $u_-(z)$ the formal series \begin{equation} u_+(z)=\frac{1}{2\pi z^{1/4}}\:\text{exp}\:(\tfrac{2}{3}z^{3/2})\sum^{\infty}_{s=0}\frac{\Gamma(3s+\frac{1}{2})}{(2s)!9^s}z^{-3s/2} \end{equation} and \begin{equation} u_-(z)=\frac{1}{2\pi z^{1/4}}\:\text{exp}\:(-\tfrac{2}{3}z^{3/2})\sum^{\infty}_{s=0}\frac{(-1)^s\Gamma(3s+\frac{1}{2})}{(2s)!9^s}\: z^{-3s/2}, \end{equation} then the asymptotic behavior of $\text{Ai}(z)$ is given by \begin{equation} \text{Ai}\;(z) \sim u_-(z), \end{equation} as $z\to\infty$ in $-\pi <\text{arg}\:z<\pi$. (Clearly, this result can not be valid in a wider sector, since $\text{Ai}(z)$ is a single-valued function and the factor multiplying the infinite series in (3) is not.) On the other hand, as $z\to\infty$ in $\frac{1}{3} \pi <\text{arg}\:z<\frac{5}{3}\pi$, we have the compound expansion \begin{equation} \text{Ai}(z)\sim u_-(z)+iu_+(z). \end{equation} In the common sector of validity, given by $\frac{1}{3}\pi < \text{arg}\:z<\pi$, either (4) or (5) can be used since the contribution of $u_+(z)$ is exponentially small compared with that of $u_-(z)$. However, in the sector $-\frac{1}{3}\pi<\text{arg}\:z<\frac{1}{3}\pi, u_+(z)$ dominates $u_-(z)$; hence it is mandatory to drop $u_+(z)$ from the asymptotic expansion of $\text{Ai}(z)$. By introducing a constant (coefficient) $C$, which is 0 for arg $z \in (-\frac{1}{3}\pi, \frac{1}{3}\pi)$ and $i$ for arg $z \in (\frac{1}{3}\pi, \frac{5}{3}\pi)$, to the asymptotic series $u_+(z)$, the two results (4) and (5) can be combined into one \begin{equation} \text{Ai}(z)\sim u_-(z)+Cu_+(z), \end{equation} as $z\to \infty$ in arg $z \in (-\pi,\frac{5}{3}\pi)$. The coefficient $C$ is called a {\it Stokes multiplier}, and is domain dependent. The discontinuous change of the coefficient $C$, when the argument of $z$ changes in a continuous manner, is known as {\it Stokes' phenomenon}. Returning to the series $u_+(z)$ and $u_-(z)$ in (2) and (3), we let $$S_+(z)=\tfrac{2}{3}\:z^{3/2}\qquad\qquad\qquad\text{and}\qquad\qquad\qquad S_-(z)=-\tfrac{2}{3}\:z^{3/2}.$$ Note that the behavior of $u_+(z)$ and $u_-(z)$ are most unequal on the curves given by \begin{equation} \text{Im}\{S_+(z)-S_-(z)\}=0, \end{equation} and they are nearly equal on the curves given by \begin{equation} \text{Re}\:\{S_+(z)-S_-(z)\}=0. \end{equation} The curves given in (7) and (8) are known, respectived, as the Stokes and anti-Stokes lines. In the case of Airy function, it is easily seen that the rays arg $z=0, \pm\frac{2}{3}\pi$ are the Stokes lines and the rays arg $z=\pm\frac{\pi}{3}, \pm\pi$ are the anti-Stokes lines. Since we are dealing with a continuous (in fact, analytic) function, it is rather unsatisfactory to have a discontinuous coefficient in the asymptotic expansion (6). In 1989, Berry wrote an innovative paper [1], in which he adopted a different interpretation of the Stokes phenomenon. In his view, the coefficient of the series $u_+(z)$ should be a continuous function of arg $z$, instead of a discontinuous constant. He gave a beautiful, although not mathematically rigorous, demonstration of this theory by truncating the series $u_-(z)$ at its optimal number of terms and the series $u_+(z)$ at its first term. Berry's theory has since become known variously as ``exponential asymptotics''or ``superasymptotics'', and has been successfully applied to several known asymptotic expansions in a mathematically rigorous manner (see, for example, Boyd [3]; Olver [5, 6]; Olde Daalhuis and Olver [4]; Wong and Zhao [7, 8]). In this lecture, we shall illustrate Berry's theory with the simple Airy function given in (1). Our approach is based on a modified version of the steepest-descent method introduced by Berry and Howls [2]. \newpage \centerline{REFERENCES} \bigskip \begin{enumerate} \item M. V. Berry, {\it Uniform asymptotic smoothing of Stokes' discontinuities}, Proc. Roy. Soc. Lond. A{\bf 422} (1989), 7 -- 21. \item M. V. Berry and C. J. Howls, {\it Hyperasymptotics for integrals with saddles}, Proc. Roy. Soc. Lond. A{\bf 434} (1991), 657 -- 675. \item W. G. C. Boyd, {\it Stieltjes transforms and the Stokes phenomenon}, Proc. Roy. Soc. Lond. A{\bf 429} (1990), 227 -- 246. \item A. B. Olde Daalhuis and F. W. J. Olver, {\it Exponentially improved asymptotic solutions of ordinary differential equations} II. {\it Irregular singularities of rank one}, Proc. Roy. Soc. Lond. A{\bf 445} (1994), 39 -- 56. \item F. W. J. Olver, {\it Uniform, exponentially improved, asymptotic expansions for the generalized exponential integrals}, SIAM J. Math. Anal. {\bf 22} (1991), 1460 -- 1474. \item F. W. J. Olver, {\it Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms}, SIAM J. Math. Anal. {\bf 22} (1991), 1475 -- 1489. \item R. Wong and Y. -Q. Zhao, {\it Smoothing of Stokes' discontinuity for the generalized Bessel function}, Proc. Roy. Soc. Lond. A{\bf 455} (1999), 1381 -- 1400. \item R. Wong and Y. -Q. Zhao, {\it Smoothing of Stokes' discontinuity for the generalized Bessel function} II, Proc. Roy. Soc. Lond. A{\bf 455} (1999), 3065 -- 3084. \end{enumerate} \end{document}