An Explicit Formula for the Coefficients of the Saddle Point Method

Abstract

Most standard textbooks about asymptotic approximations of integrals do not give explicit formulas for the coefficients of the asymptotic methods of Laplace and saddle point. In these techniques, those coefficients arise as the Taylor coefficients of a function defined in an implicit form, and the coefficients are not given by a closed algebraic formula. Despite this fact, we can extract from the literature some formulas of varying degrees of explicitness for those coefficients: Perron’s method (in Sitzungsber. Bayr. Akad. Wissensch. (Munch. Ber.), 191–219, 1917) offers an explicit computation in terms of the derivatives of an explicit function; in (de Bruijn, Asymptotic Methods in Analysis. Dover, New York, 1950) we can find a similar formula for the Laplace method which uses derivatives of an explicit function. Dingle (in Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1973) gives the coefficients of the saddle point method in terms of a contour integral. Perron’s method is rediscovered in (Campbell et al., Stud. Appl. Math. 77:151–172, 1987), but they also go farther and compute the above mentioned derivatives by means of a recurrence. The most recent contribution is (Wojdylo, SIAM Rev. 48(1):76–96, 2006), which rediscovers the Campbell, Froman and Walles’ formula and rewrites it in terms of Bell polynomials (in the Laplace method) using new ideas of combinatorial analysis which efficiently simplify and systematize the computations. In this paper we continue the research line of these authors. We combine the more systematic version of the saddle point method introduced in (Lopez et al., J. Math. Anal. Appl. 354(1):347–359, 2009) with Wojdylo’s idea to derive a new and more explicit formula for the coefficients of the saddle point method, similar to Wojdylo’s formula for the coefficients of the Laplace method. As an example, we show the application of this formula to the Bessel function.