# Stationary phase, method of the

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

where , , , is a large parameter, is a bounded domain, the function (the phase) is real, the function is complex, and . If , i.e. has compact support, and the phase does not have stationary points (i.e. points at which ) on , , then , for all as . Therefore, when , the points of stationary phase and the boundary give the essential contribution to the asymptotics of the integral (*). The integrals  are called the contributions from the isolated stationary point and the boundary, respectively, where , near the point and does not contain any other stationary points, and in a certain neighbourhood of the boundary. For , :

1) , if ;

2)  if is an interior point of and , .

Detailed research has been carried out in the case where , the phase has a finite number of stationary points, all of finite multiplicity, and the function has zeros of finite multiplicity at these points and at the end-points of an interval . Asymptotic expansions have been obtained. The case where the functions and have power singularities has also been studied: for example, , , where , are smooth functions when , , .

Let , and let be a non-degenerate stationary point (i.e. ). The contribution from the point is then equal to  where is the signature of the matrix . There is also an asymptotic series for (for the formulas of the contribution in the case of a smooth boundary, see ).

If is a stationary point of finite multiplicity, then (see ) where are rational numbers, . Degenerate stationary points have been studied, cf. , .

Studies have been made on the case where the phase depends on a real parameter , and for small has two close non-degenerate stationary points. In this case, the asymptotics of the integral can be expressed in terms of Airy functions (see , ). The method of the stationary phase has an operator variant: , where is the infinitesimal operator of the strongly-continuous group of operators bounded on the axis , acting on a Banach space , and , are smooth functions with values in . If the functions are analytic, then the method of the stationary phase is a particular case of the saddle point method.

How to Cite This Entry:
Stationary phase, method of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stationary_phase,_method_of_the&oldid=16013
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article