View source for Stationary phase, method of the

A method for calculating the asymptotics of integrals of rapidly-oscillating functions:

<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/s087/s087270/s0872701.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/s087/s087270/s0872702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872704.png" />, is a large parameter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872705.png" /> is a bounded domain, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872706.png" /> (the phase) is real, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872707.png" /> is complex, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872708.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s0872709.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727010.png" /> has compact support, and the phase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727011.png" /> does not have stationary points (i.e. points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727012.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727015.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727016.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727017.png" />. Therefore, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727018.png" />, the points of stationary phase and the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727019.png" /> give the essential contribution to the asymptotics of the integral (*). The integrals

<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/s087/s087270/s08727020.png" /></td> </tr></table>

<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/s087/s087270/s08727021.png" /></td> </tr></table>

are called the contributions from the isolated stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727022.png" /> and the boundary, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727024.png" /> near the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727026.png" /> does not contain any other stationary points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727028.png" /> in a certain neighbourhood of the boundary. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727030.png" />:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727031.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727032.png" />;

2)

<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/s087/s087270/s08727033.png" /></td> </tr></table>

<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/s087/s087270/s08727034.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727035.png" /> is an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727038.png" />.

Detailed research has been carried out in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727039.png" />, the phase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727040.png" /> has a finite number of stationary points, all of finite multiplicity, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727041.png" /> has zeros of finite multiplicity at these points and at the end-points of an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727042.png" />. Asymptotic expansions have been obtained. The case where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727044.png" /> have power singularities has also been studied: for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727048.png" /> are smooth functions when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727051.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727052.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727053.png" /> be a non-degenerate stationary point (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727054.png" />). The contribution from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727055.png" /> is then equal to

<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/s087/s087270/s08727056.png" /></td> </tr></table>

<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/s087/s087270/s08727057.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727058.png" /> is the [[Signature|signature]] of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727059.png" />. There is also an asymptotic series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727060.png" /> (for the formulas of the contribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727061.png" /> in the case of a smooth boundary, see [[#References|[5]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727062.png" /> is a stationary point of finite multiplicity, then (see [[#References|[6]]])

<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/s087/s087270/s08727063.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727064.png" /> are rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727065.png" />. Degenerate stationary points have been studied, cf. [[#References|[3]]], [[#References|[4]]].

Studies have been made on the case where the phase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727066.png" /> depends on a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727067.png" />, and for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727068.png" /> has two close non-degenerate stationary points. In this case, the asymptotics of the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727069.png" /> can be expressed in terms of [[Airy functions|Airy functions]] (see [[#References|[5]]], [[#References|[10]]]). The method of the stationary phase has an operator variant: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727071.png" /> is the infinitesimal operator of the strongly-continuous group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727072.png" /> of operators bounded on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727073.png" />, acting on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727074.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727076.png" /> are smooth functions with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087270/s08727077.png" /> [[#References|[9]]]. If the functions are analytic, then the method of the stationary phase is a particular case of the [[Saddle point method|saddle point method]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Thomson,   ''Philos. Mag.'' , '''23'''  (1887)  pp. 252–255</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">  E.Ya. Rieksteyn'sh,   "Asymptotic expansions of integrals" , '''1–2''' , Riga  (1974–1977)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.W.J. Olver,   "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.V. Fedoryuk,   "The method of steepest descent" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.F. Atiyah,   "Resolution of singularities and division of distributions"  ''Comm. Pure Appl. Math.'' , '''23''' :  2  (1970)  pp. 145–150</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.I. Arnol'd,   "Remarks on the stationary phase method and Coxeter numbers"  ''Russian Math. Surveys'' , '''28''' :  5  (1973)  pp. 19–48  ''Uspekhi Mat. Nauk'' , '''28''' :  5  (1973)  pp. 17–44</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.N. Varchenko,   "Newton polyhedra and estimation of oscillating integrals"  ''Funct. Anal. Appl'' , '''10''' :  3  (1976)  pp. 175–196  ''Funktsional. Anal. i Prilozhen.'' , '''10''' :  3  (1976)  pp. 13–38</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.P. Maslov,   M.V. Fedoryuk,   "Semi-classical approximation in quantum mechanics" , Reidel  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.V. Fedoryuk,   "Asymptotics. Integrals and series" , Moscow  (1987)  (In Russian)</TD></TR></table>



====Comments====
An integral of the form (*) is a special case of a so-called oscillatory integral, or [[Fourier integral operator|Fourier integral operator]], cf. also [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Wong,   "Asymptotic approximations of integrals" , Acad. Press  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. §7.7</TD></TR></table>