Method of Stationary Phase for Single Integrals
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Method of Stationary Phase for Single Integrals book
A rigorous mathematical justification of the principle of stationary phase may be found in Erdelyi. In this chapter, the authors take an approach that is heuristic but nevertheless produces correct results. This heuristic approach was used for the first time by Lord Kelvin (Thomson 1887), who applied it to a problem in water wave propagation; much more recently it has been applied to diffraction problems of imaging by van Kampen (1949). To obtain the asymptotic contribution of a non-stationary end-point we may follow a similar procedure as in the previous two cases. But it turns out to be just as easy to integrate by parts. The asymptotic formulae become invalid when the stationary points are not isolated from one another or from end-points. The derivation of Felsen and Marcuwitz's uniform asymptotic result is complicated. In diffraction theory two nearby stationary points are associated with a caustic of rays.