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A method for computing the asymptotic expansion of integrals of the form
 
A method for computing the asymptotic expansion of integrals of the form
  
<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/s/s083/s083070/s0830701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
F( \lambda )  = \int\limits _  \gamma  f( z) e ^ {\lambda S( z) }  dz,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830703.png" /> is a large parameter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830704.png" /> is a contour in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830705.png" />-plane, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830707.png" /> are holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830708.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s0830709.png" />. The zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307010.png" /> are called the saddle points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307011.png" />. The essence of the method is as follows. The contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307012.png" /> is deformed to a contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307013.png" /> with the same end-points and lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307014.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307015.png" /> is attained only at the saddle points or at the ends of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307016.png" /> (the contour of steepest descent). The asymptotics of the integral (*) along the path of steepest descent are calculated by means of the [[Laplace method|Laplace method]] and are equal to the sum of the contributions from the saddle points. The contribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307017.png" /> from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307018.png" /> is an integral of the form of (*) taken over a small arc of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307019.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307021.png" /> is an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307023.png" /> is a saddle point with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307024.png" />, then
+
where $  \lambda > 0 $,  
 +
$  \lambda \rightarrow + \infty $
 +
is a large parameter, $  \gamma $
 +
is a contour in the complex $  z $-
 +
plane, and the functions $  f( z) $
 +
and $  S( z) $
 +
are holomorphic in a domain $  D $
 +
containing $  \gamma $.  
 +
The zeros of $  S  ^  \prime  ( z) $
 +
are called the saddle points of $  S( z) $.  
 +
The essence of the method is as follows. The contour $  \gamma $
 +
is deformed to a contour $  \widetilde \gamma  $
 +
with the same end-points and lying in $  D $
 +
and such that $  \max _ {z \in \widetilde \gamma  }    \mathop{\rm Re}  S( z) $
 +
is attained only at the saddle points or at the ends of $  \widetilde \gamma  $(
 +
the contour of steepest descent). The asymptotics of the integral (*) along the path of steepest descent are calculated by means of the [[Laplace method|Laplace method]] and are equal to the sum of the contributions from the saddle points. The contribution $  V _ {z _ {0}  } ( \lambda ) $
 +
from the point $  z _ {0} $
 +
is an integral of the form of (*) taken over a small arc of $  \widetilde \gamma  $
 +
containing the point $  z _ {0} $.  
 +
If $  z _ {0} $
 +
is an interior point of $  \widetilde \gamma  $
 +
and $  z _ {0} $
 +
is a saddle point with $  S  ^ {\prime\prime} ( z _ {0} ) \neq 0 $,  
 +
then
  
<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/s/s083/s083070/s08307025.png" /></td> </tr></table>
+
$$
 +
V _ {z _ {0}  } ( \lambda )  = \sqrt {-  
 +
\frac{2 \pi }{\lambda S  ^ {\prime\prime}
 +
( z _ {0} ) }
 +
} e ^ {\lambda S( z _ {0} ) } [ f( z _ {0} ) + O( \lambda  ^ {-} 1 )].
 +
$$
  
 
The contour of steepest descent has a minimax property; on it,
 
The contour of steepest descent has a minimax property; on it,
  
<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/s/s083/s083070/s08307026.png" /></td> </tr></table>
+
$$
 +
\min _ {\gamma  ^  \prime  }  \max _ {z \in \gamma  ^  \prime  }  \mathop{\rm Re}
 +
S( z)
 +
$$
  
is attained, where the minimum is taken over all contours <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307027.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307028.png" /> having the same end-points as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307029.png" />. The main difficulty in using the method is to select the saddle points, i.e. to choose the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307030.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307031.png" />.
+
is attained, where the minimum is taken over all contours $  \gamma  ^  \prime  $
 +
lying in $  D $
 +
having the same end-points as $  \gamma $.  
 +
The main difficulty in using the method is to select the saddle points, i.e. to choose the $  \widetilde \gamma  $
 +
corresponding to $  \gamma $.
  
 
The method is due to P. Debye [[#References|[1]]], although the ideas in the method were suggested earlier by B. Riemann [[#References|[2]]]. See [[#References|[3]]]–[[#References|[9]]] for the calculation of the contributions from the saddle points and from the end-points of the contour.
 
The method is due to P. Debye [[#References|[1]]], although the ideas in the method were suggested earlier by B. Riemann [[#References|[2]]]. See [[#References|[3]]]–[[#References|[9]]] for the calculation of the contributions from the saddle points and from the end-points of the contour.
Line 17: Line 66:
 
The method is in essence the only method for calculating the asymptotic expansions of integrals of the form (*). It can be used to derive the asymptotic expansions for Laplace, Fourier and Mellin transforms, as well as for transforms of exponentials of polynomials and many special functions.
 
The method is in essence the only method for calculating the asymptotic expansions of integrals of the form (*). It can be used to derive the asymptotic expansions for Laplace, Fourier and Mellin transforms, as well as for transforms of exponentials of polynomials and many special functions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307032.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307033.png" /> be a bounded manifold with boundary of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307034.png" /> and of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307035.png" />, let functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307037.png" /> be holomorphic in a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307038.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307039.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307040.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307041.png" /> is attained at a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307042.png" /> which is an interior point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307043.png" /> and a non-singular saddle point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307044.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307045.png" />. Then the contribution from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083070/s08307046.png" /> is
+
Let $  z \in \mathbf C  ^ {n} $,  
 +
let $  \gamma $
 +
be a bounded manifold with boundary of dimension $  n $
 +
and of class $  C  ^  \infty  $,  
 +
let functions $  f( z) $
 +
and $  S( z) $
 +
be holomorphic in a certain domain $  D $
 +
containing $  \gamma $,  
 +
and let $  dz = dz _ {1} \dots dz _ {n} $.  
 +
Suppose that $  \max _ {z \in \gamma }    \mathop{\rm Re}  S( z) $
 +
is attained at a single point $  z  ^ {0} $
 +
which is an interior point for $  \gamma $
 +
and a non-singular saddle point for $  S( z) $,  
 +
i.e. $  \Delta _ {S} ( z  ^ {0} ) \equiv  \mathop{\rm det}  S  ^ {\prime\prime} ( z  ^ {0} ) \neq 0 $.  
 +
Then the contribution from $  z  ^ {0} $
 +
is
  
<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/s/s083/s083070/s08307047.png" /></td> </tr></table>
+
$$
 +
F( \lambda )  = \left (
 +
\frac{2 \pi } \lambda
 +
\right )  ^ {n/2} (- \Delta _ {S} ( z
 +
^ {0} ))  ^ {-} 1/2 e ^ {\lambda S( z  ^ {0} ) } [ f( z  ^ {0} ) + O(
 +
\lambda  ^ {-} 1 )].
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Debye,  "Näherungsformeln für die Zylinderfunktionen für grosse Werte des Arguments und unbeschränkt veranderliche Werte des Index"  ''Math. Ann.'' , '''67'''  (1909)  pp. 535–558</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Riemann,  "Mathematische Werke" , Dover, reprint  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Erdélyi,  "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.G. de Bruijn,  "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Evgrafov,  "Asymptotic estimates and entire functions" , Gordon &amp; Breach  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.T. Copson,  "Asymptotic expansions" , Cambridge Univ. Press  (1965)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  F.W.J. Olver,  "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.Ya. Riekstyn'sh,  "Asymptotic expansions of integrals" , '''1–2''' , Riga  (1974–1977)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.V. Fedoryuk,  "The saddle-point method" , Moscow  (1977)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Debye,  "Näherungsformeln für die Zylinderfunktionen für grosse Werte des Arguments und unbeschränkt veranderliche Werte des Index"  ''Math. Ann.'' , '''67'''  (1909)  pp. 535–558</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Riemann,  "Mathematische Werke" , Dover, reprint  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Erdélyi,  "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.G. de Bruijn,  "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Evgrafov,  "Asymptotic estimates and entire functions" , Gordon &amp; Breach  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.T. Copson,  "Asymptotic expansions" , Cambridge Univ. Press  (1965)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  F.W.J. Olver,  "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.Ya. Riekstyn'sh,  "Asymptotic expansions of integrals" , '''1–2''' , Riga  (1974–1977)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.V. Fedoryuk,  "The saddle-point method" , Moscow  (1977)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Wong,  "Asymptotic approximations of integrals" , Acad. Press  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Wong,  "Asymptotic approximations of integrals" , Acad. Press  (1989)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


A method for computing the asymptotic expansion of integrals of the form

$$ \tag{* } F( \lambda ) = \int\limits _ \gamma f( z) e ^ {\lambda S( z) } dz, $$

where $ \lambda > 0 $, $ \lambda \rightarrow + \infty $ is a large parameter, $ \gamma $ is a contour in the complex $ z $- plane, and the functions $ f( z) $ and $ S( z) $ are holomorphic in a domain $ D $ containing $ \gamma $. The zeros of $ S ^ \prime ( z) $ are called the saddle points of $ S( z) $. The essence of the method is as follows. The contour $ \gamma $ is deformed to a contour $ \widetilde \gamma $ with the same end-points and lying in $ D $ and such that $ \max _ {z \in \widetilde \gamma } \mathop{\rm Re} S( z) $ is attained only at the saddle points or at the ends of $ \widetilde \gamma $( the contour of steepest descent). The asymptotics of the integral (*) along the path of steepest descent are calculated by means of the Laplace method and are equal to the sum of the contributions from the saddle points. The contribution $ V _ {z _ {0} } ( \lambda ) $ from the point $ z _ {0} $ is an integral of the form of (*) taken over a small arc of $ \widetilde \gamma $ containing the point $ z _ {0} $. If $ z _ {0} $ is an interior point of $ \widetilde \gamma $ and $ z _ {0} $ is a saddle point with $ S ^ {\prime\prime} ( z _ {0} ) \neq 0 $, then

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

The contour of steepest descent has a minimax property; on it,

$$ \min _ {\gamma ^ \prime } \max _ {z \in \gamma ^ \prime } \mathop{\rm Re} S( z) $$

is attained, where the minimum is taken over all contours $ \gamma ^ \prime $ lying in $ D $ having the same end-points as $ \gamma $. The main difficulty in using the method is to select the saddle points, i.e. to choose the $ \widetilde \gamma $ corresponding to $ \gamma $.

The method is due to P. Debye [1], although the ideas in the method were suggested earlier by B. Riemann [2]. See [3][9] for the calculation of the contributions from the saddle points and from the end-points of the contour.

The method is in essence the only method for calculating the asymptotic expansions of integrals of the form (*). It can be used to derive the asymptotic expansions for Laplace, Fourier and Mellin transforms, as well as for transforms of exponentials of polynomials and many special functions.

Let $ z \in \mathbf C ^ {n} $, let $ \gamma $ be a bounded manifold with boundary of dimension $ n $ and of class $ C ^ \infty $, let functions $ f( z) $ and $ S( z) $ be holomorphic in a certain domain $ D $ containing $ \gamma $, and let $ dz = dz _ {1} \dots dz _ {n} $. Suppose that $ \max _ {z \in \gamma } \mathop{\rm Re} S( z) $ is attained at a single point $ z ^ {0} $ which is an interior point for $ \gamma $ and a non-singular saddle point for $ S( z) $, i.e. $ \Delta _ {S} ( z ^ {0} ) \equiv \mathop{\rm det} S ^ {\prime\prime} ( z ^ {0} ) \neq 0 $. Then the contribution from $ z ^ {0} $ is

$$ F( \lambda ) = \left ( \frac{2 \pi } \lambda \right ) ^ {n/2} (- \Delta _ {S} ( z ^ {0} )) ^ {-} 1/2 e ^ {\lambda S( z ^ {0} ) } [ f( z ^ {0} ) + O( \lambda ^ {-} 1 )]. $$

References

[1] P. Debye, "Näherungsformeln für die Zylinderfunktionen für grosse Werte des Arguments und unbeschränkt veranderliche Werte des Index" Math. Ann. , 67 (1909) pp. 535–558
[2] B. Riemann, "Mathematische Werke" , Dover, reprint (1953)
[3] A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
[4] N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[5] M.A. Evgrafov, "Asymptotic estimates and entire functions" , Gordon & Breach (1962) (Translated from Russian)
[6] E.T. Copson, "Asymptotic expansions" , Cambridge Univ. Press (1965)
[7] F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974)
[8] E.Ya. Riekstyn'sh, "Asymptotic expansions of integrals" , 1–2 , Riga (1974–1977) (In Russian)
[9] M.V. Fedoryuk, "The saddle-point method" , Moscow (1977) (In Russian)

Comments

References

[a1] R. Wong, "Asymptotic approximations of integrals" , Acad. Press (1989)
How to Cite This Entry:
Saddle point method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saddle_point_method&oldid=48605
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article