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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

Asymptotic Expansion Analytic Continuation Meromorphic Function Irreducible Component Arithmetic Progression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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