Elementary integrals and the resolution of singularities of the phase

  • V. I. Arnold
  • S. M. Gusein-Zade
  • A. N. Varchenko
Part of the Monographs in Mathematics book series (MMA, volume 83)

Abstract

In this chapter we shall study the asymptotics of an oscillatory integral, the phase of which is a monomial. We shall indicate the connection between the asymptotics of an oscillatory integral and the poles of the meromorphic function
$$F(\lambda)=\smallint{f^\lambda}(x)\Phi(x)dx,$$
, Where ƒ is the phase, and ϕ is the amplitude of the oscillatory integral. We shall introduce the discrete characteristics of the resolution of the singularity of a critical point of the phase: the weight of the resolution and the multiplicity set. We shall describe the connection between these characteristics and the basic characteristics of the asymptotic behaviour of the oscillatory integral: the oscillation index, its multiplicity and the index set.

Keywords

Manifold 

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Copyright information

© Birkhäuser Boston, Inc. 1988

Authors and Affiliations

  • V. I. Arnold
  • S. M. Gusein-Zade
  • A. N. Varchenko

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