View source for Saddle point method

<!--
s0830701.png
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/S083/S.0803070 Saddle point method
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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|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 [[#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 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====
<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====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Wong,   "Asymptotic approximations of integrals" , Acad. Press  (1989)</TD></TR></table>