On the asymptotics of some Pearcey-type integrals

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The author discusses the asymptotic behaviour of the Pearcey-type integralI'alpha(X, Y)=2 (sup) approximately=0 ualpha +1 exp(i(u4+Xu2))Jalpha(Yu) du for -1( alpha (5/2, where Jalpha is a Bessel function, as X to +or- infinity , Y fixed, as Y to infinity , X fixed, and as Y= rho (2/3 mod X mod )32/, X to - infinity , rho fixed. The case alpha =- 1/2 gives the classical Pearcey integral whose asymptotics has been investigated recently by Kaminski (1989) and Paris (1991). In the case alpha =0, I'alpha(X, Y) as a function of Y>or=0 represents the radial part of the impulse-response function describing the image formation in high resolution electron microscopes at normalized defocus X. He uses the approach of Paris by representing I'alpha(X, Y) in terms of Weber parabolic cylinder functions, and he augments this approach by invoking the Chester-Friedman-Ursell method (1957) to obtain the leading asymptotics of I'alpha(X, Y) around the caustic Y2=(2/3 mod X mod )3, X to - infinity.
Original languageEnglish
Pages (from-to)L823-L831
Number of pages9
JournalJournal of Physics A: Mathematical and General
Issue number13
Publication statusPublished - 1992


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