# 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)