Laplace method

of asymptotic estimation

A method for determining the asymptotic behaviour as of Laplace integrals

(1)

where is a finite interval, is a real-valued function and is a complex-valued function, both sufficiently smooth for . The asymptotic behaviour of is the sum of the contributions from points at which is attained, if the number of these points is assumed to be finite.

1) If a maximum is attained at and if , then the contribution from the point in the asymptotic behaviour of the integral (1) is equal to

2) If a maximum is attained at an interior point of the interval and , then its contribution equals

This formula was obtained by P.S. Laplace [1]. The case when and have zeros of finite multiplicity at maximum points of has been completely investigated, and asymptotic expansions have been obtained (see [2]–[8]). The Laplace method can also be extended to the case of a contour in the complex plane (see Saddle point method).

Let be a bounded domain in and suppose that the maximal of in the closure of is attained only at an interior point , where is a non-degenerate stationary point of . Then

In this case, asymptotic expansions for have also been obtained. All the formulas given above hold for complex , , . There are also modifications of the Laplace method for the case when the dependence on the parameter is more complicated (see [4], [8]):

References

[1]	P.S. Laplace, "Essai philosophique sur les probabilités" , Oeuvres complètes , 7 , Gauthier-Villars (1886)
[2]	A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
[3]	N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[4]	M.A. Evgrafov, "Asymptotic estimates and entire functions" , Gordon & Breach (1961) (Translated from Russian)
[5]	E.T. Copson, "Asymptotic expansions" , Cambridge Univ. Press (1965)
[6]	F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974)
[7]	E. Riekstyn'sh, "Asymptotic expansions of integrals" , 1 , Riga (1974) (In Russian)
[8]	M.V. Fedoryuk, "The method of steepest descent" , Moscow (1977) (In Russian)

References

[a1]	N. Bleistein, R.A. Handelsman, "Asymptotic expansions of integrals" , Holt, Rinehart & Winston (1975) pp. Chapt. 5