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- W242931454 abstract "The aim of this paper is to calculate the principal term of the asymptotic expansion of an oscillating integral in a neighborhood of a singular critical point of the phase in terms of Newton's diagram of the phase expansion in a Taylor series in a neighborhood of this critical point. The main result of this paper consists in the fact that this principal term is determined by the point of intersection of Newton's diagram with the diagonal of the coordinate octant (under the conditions formulated below). Under these conditions the arithmetic progressions to which the indices of all the terms of the asymptotic expansion belong will depend only on Newton's diagram of the phase function. In this paper we specify the form of these progressions in terms of Newton's diagram. The obtained formulas confirm the hypothesis of V. I. Arnol'd that all appropriate discrete invariants of an analytic function can be expressed in a simple manner in terms of Newton's diagram for almost all the functions with a given Newton diagram (see [ii and 13]). We calculate below the indices of the principal terms of the asymptotic expansion for all the phase functions classified in [16] (in two cases our theorems yield an inequality for the index of the principal term). We present an example that refutes the hypothesis of semicontinuity of the index of the principal term of the asymptotic expansion." @default.
- W242931454 created "2016-06-24" @default.
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- W242931454 date "1976-01-01" @default.
- W242931454 modified "2023-09-27" @default.
- W242931454 title "0.i. Newton's Diagram. Let NC RC R+ be the sets of all nonnegative integers, all nonnegative real numbers, and all real numbers respectively. Let K~ N ~. Definition. Newton's polyhedron of a set K is defined by the convex hull in R k of the" @default.
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