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''of asymptotic estimation''
 
''of asymptotic estimation''
  
A method for determining the asymptotic behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575001.png" /> of Laplace integrals
+
A method for determining the asymptotic behaviour as $  0 < \lambda \rightarrow + \infty $
 +
of Laplace integrals
 +
 
 +
$$ \tag{1 }
 +
F ( \lambda )  =  \int\limits _  \Omega
 +
f ( x) e ^ {\lambda S ( x) }  d x ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575003.png" /> is a finite interval, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575004.png" /> is a real-valued function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575005.png" /> is a complex-valued function, both sufficiently smooth for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575006.png" />. The asymptotic behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575007.png" /> is the sum of the contributions from points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575008.png" /> 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
  
1) If a maximum is attained at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l0575009.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750010.png" />, then the contribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750011.png" /> from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750012.png" /> in the asymptotic behaviour of the integral (1) is equal to
+
$$
 +
V _ {a} ( \lambda ) =  -
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750013.png" /></td> </tr></table>
+
\frac{f ( a) + O ( \lambda  ^ {-} 1 ) }{\lambda S ^ { \prime } ( a) }
  
2) If a maximum is attained at an interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750014.png" /> of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750016.png" />, then its contribution equals
+
e ^ {\lambda S ( a) } .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750017.png" /></td> </tr></table>
+
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
  
This formula was obtained by P.S. Laplace [[#References|[1]]]. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750019.png" /> have zeros of finite multiplicity at maximum points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750020.png" /> has been completely investigated, and asymptotic expansions have been obtained (see [[#References|[2]]]–[[#References|[8]]]). The Laplace method can also be extended to the case of a contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750021.png" /> in the complex plane (see [[Saddle point method|Saddle point method]]).
+
$$
 +
V _ {x  ^ {0}  } ( \lambda )  = \
 +
\sqrt {-
 +
\frac{2 \pi }{\lambda S ^ { \prime\prime } ( x  ^ {0} ) }
 +
}
 +
[ f ( x  ^ {0} ) + O ( \lambda  ^ {-} 1 ) ] e ^
 +
{\lambda S ( x  ^ {0} ) } .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750022.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750023.png" /> and suppose that the maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750025.png" /> in the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750026.png" /> is attained only at an interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750028.png" /> is a non-degenerate stationary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750029.png" />. Then
+
This formula was obtained by P.S. Laplace [[#References|[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 [[#References|[2]]]–[[#References|[8]]]). The Laplace method can also be extended to the case of a contour  $  \Omega $
 +
in the complex plane (see [[Saddle point method|Saddle point method]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750030.png" /></td> </tr></table>
+
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
  
In this case, asymptotic expansions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750031.png" /> have also been obtained. All the formulas given above hold for complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750034.png" />. There are also modifications of the Laplace method for the case when the dependence on the parameter is more complicated (see [[#References|[4]]], [[#References|[8]]]):
+
$$
 +
F ( \lambda )  = \left (
 +
\frac{2 \pi } \lambda
 +
\right )  ^ {n/2}
 +
|  \mathop{\rm det}  S _ {xx} ^ { \prime\prime } ( x  ^ {0} ) |  ^ {-} 1/2
 +
[ f ( x  ^ {0} ) + O ( \lambda  ^ {-} 1 ) ]
 +
e ^ {\lambda S ( x  ^ {0} ) } .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057500/l05750035.png" /></td> </tr></table>
+
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 [[#References|[4]]], [[#References|[8]]]):
 +
 
 +
$$
 +
F ( \lambda )  = \int\limits _ {\Omega ( \lambda ) }
 +
f ( x , \lambda ) e ^ {S ( x , \lambda ) }  d x .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Laplace,  "Essai philosophique sur les probabilités" , ''Oeuvres complètes'' , '''7''' , Gauthier-Villars  (1886)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Erdélyi,  "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. de Bruijn,  "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Evgrafov,  "Asymptotic estimates and entire functions" , Gordon &amp; Breach  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.T. Copson,  "Asymptotic expansions" , Cambridge Univ. Press  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  F.W.J. Olver,  "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Riekstyn'sh,  "Asymptotic expansions of integrals" , '''1''' , Riga  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.V. Fedoryuk,  "The method of steepest descent" , Moscow  (1977)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Laplace,  "Essai philosophique sur les probabilités" , ''Oeuvres complètes'' , '''7''' , Gauthier-Villars  (1886)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Erdélyi,  "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. de Bruijn,  "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Evgrafov,  "Asymptotic estimates and entire functions" , Gordon &amp; Breach  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.T. Copson,  "Asymptotic expansions" , Cambridge Univ. Press  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  F.W.J. Olver,  "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Riekstyn'sh,  "Asymptotic expansions of integrals" , '''1''' , Riga  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.V. Fedoryuk,  "The method of steepest descent" , Moscow  (1977)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bleistein,  R.A. Handelsman,  "Asymptotic expansions of integrals" , Holt, Rinehart &amp; Winston  (1975)  pp. Chapt. 5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bleistein,  R.A. Handelsman,  "Asymptotic expansions of integrals" , Holt, Rinehart &amp; Winston  (1975)  pp. Chapt. 5</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


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