GENERALIZED FRESNEL INTEGRALS AS OSCILLATORY INTEGRALS WITH POSITIVE REAL POWER PHASE FUNCTIONS AND APPLICATIONS TO ASYMPTOTIC EXPANSIONS

Toshio Nagano, Naoya Miyazaki

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we first generalize the Fresnel integrals by changing of a path for integration in the proof of the Fresnel integrals by Cauchy’s integral theorem. Next, according to oscillatory integral, we also obtain further generalization of the extended Fresnel integrals. Moreover by using this result, we have an asymptotic expansion of an oscillatory integral with a positive real parameter, for a phase function with a degenerate critical point expressed by positive real power, including a moderate oscillation, and for a suitable amplitude function. This result gives a finer extension of the stationary phase method in one variable, which is known as a method for an asymptotic expansion of an oscillatory integral of a phase function with a non-degenerate critical point.

MSC Codes Primary 42B20, Secondary 41A60, 33B20

Original languageEnglish
JournalUnknown Journal
Publication statusPublished - 2020 May 24

Keywords

  • Asymptotic expansion
  • Fresnel integral
  • Oscillatory integral
  • Stationary phase method

ASJC Scopus subject areas

  • General

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