A systematization of the saddle point method. Application to the Airy and Hankel functions

Abstract

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function H n (z) for large n and z are given as non-trivial examples.

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

“…The key point is the factorization of the integrand in in the form where we have defined the remainder function As well as this remainder function, the functions e − xf ( t ) and have a Taylor expansion at t = t 0 : for a certain r > 0, where A n ( x ) are the Taylor coefficients of exp(− xf ( t )) at t = t 0 and c n ( x ) can be written in terms of A n ( x ) [2]: It is shown in [2] that as x →∞. As well as in the standard method of “saddle point near a pole,” we assume that g ( t ) has a pole at t 1 = t 0 − i λ (both coalesce when λ→ 0).…”

“…When the interval ( a , b ) is not contained in the disk of convergence | t − t 0 | < r of the Taylor series in the right‐hand side of and , then may not be true, but still, holds ([1], [2]).…”

“…The reason is that the standard form of the integral must encode the asymptotic feature of the integral: coalescence of saddles, or coalescence of poles and saddles, etc, … and then the change of variables is more cumbersome. In this paper, we show that the modification of the Laplace’s and saddle point methods introduced in [1] and [2] that makes those methods more systematic with universal algebraically closed formulas for the coefficients can be extended to the uniform method “saddle point near a pole” ([3], chap. 7, sec.…”

“… In recent works [1] and [2], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions.…”

“…Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [1] and [2] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions).…”