View source for Saddle point method

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>

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

<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>

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

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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" />.

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 <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

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====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>