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) |
Comments
References
| [a1] | N. Bleistein, R.A. Handelsman, "Asymptotic expansions of integrals" , Holt, Rinehart & Winston (1975) pp. Chapt. 5 |
Laplace method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_method&oldid=47580