*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) |

#### Comments

#### References

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

**How to Cite This Entry:**

Laplace method. *Encyclopedia of Mathematics.* URL: http://encyclopediaofmath.org/index.php?title=Laplace_method&oldid=47580

This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article