On a method of asymptotic evaluation of multiple integrals

Abstract

In this paper, some of the formal arguments given by Jones and Kline [<italic>J. Math. Phys.</italic>, v. 37, 1958, pp. 1-28] are made rigorous. In particular, the reduction procedure of a multiple oscillatory integral to a one-dimensional Fourier transform is justified, and a Taylor-type theorem with remainder is proved for the Dirac <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-function. The analyticity condition of Jones and Kline is now replaced by infinite differentiability. Connections with the asymptotic expansions of Jeanquartier and Malgrange are also discussed.