Laplace method

of asymptotic estimation

A method for determining the asymptotic behaviour as $ 0 < \lambda \rightarrow + \infty $ of Laplace integrals

$$ \tag{1 } F ( \lambda ) = \int\limits _ \Omega f ( x) e ^ {\lambda S ( x) } d x , $$

where $ \Omega = [ a , b ] $ is a finite interval, $ S $ is a real-valued function and $ f $ is a complex-valued function, both sufficiently smooth for $ x \in \Omega $. The asymptotic behaviour of $ F ( \lambda ) $ is the sum of the contributions from points at which $ \max _ {x \in \Omega } S ( x) $ is attained, if the number of these points is assumed to be finite.

1) If a maximum is attained at $ x = a $ and if $ S ^ { \prime } ( a) \neq 0 $, then the contribution $ V _ {a} ( \lambda ) $ from the point $ a $ in the asymptotic behaviour of the integral (1) is equal to

$$ V _ {a} ( \lambda ) = - \frac{f ( a) + O ( \lambda ^ {-} 1 ) }{\lambda S ^ { \prime } ( a) } e ^ {\lambda S ( a) } . $$

2) If a maximum is attained at an interior point $ x ^ {0} $ of the interval $ \Omega $ and $ S ^ { \prime\prime } ( x ^ {0} ) \neq 0 $, then its contribution equals

$$ V _ {x ^ {0} } ( \lambda ) = \ \sqrt {- \frac{2 \pi }{\lambda S ^ { \prime\prime } ( x ^ {0} ) } } [ f ( x ^ {0} ) + O ( \lambda ^ {-} 1 ) ] e ^ {\lambda S ( x ^ {0} ) } . $$

This formula was obtained by P.S. Laplace [1]. The case when $ f ( x) $ and $ S ^ { \prime } ( x) $ have zeros of finite multiplicity at maximum points of $ S $ 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 $ \Omega $ in the complex plane (see Saddle point method).

Let $ \Omega $ be a bounded domain in $ \mathbf R _ {x} ^ {n} $ and suppose that the maximal $ m $ of $ S ( x) $ in the closure of $ \Omega $ is attained only at an interior point $ x ^ {0} $, where $ x ^ {0} $ is a non-degenerate stationary point of $ S $. Then

$$ F ( \lambda ) = \left ( \frac{2 \pi } \lambda \right ) ^ {n/2} | \mathop{\rm det} S _ {xx} ^ { \prime\prime } ( x ^ {0} ) | ^ {-} 1/2 [ f ( x ^ {0} ) + O ( \lambda ^ {-} 1 ) ] e ^ {\lambda S ( x ^ {0} ) } . $$

In this case, asymptotic expansions for $ F ( \lambda ) $ have also been obtained. All the formulas given above hold for complex $ \lambda $, $ | \lambda | \rightarrow \infty $, $ | \mathop{\rm arg} \lambda | \leq \pi / 2 - \epsilon $. There are also modifications of the Laplace method for the case when the dependence on the parameter is more complicated (see [4], [8]):

$$ F ( \lambda ) = \int\limits _ {\Omega ( \lambda ) } f ( x , \lambda ) e ^ {S ( x , \lambda ) } d x . $$

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