Generalized Fresnel integrals as oscillatory integrals with positive real power phase functions and applications to asymptotic expansions

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.

arXiv admin note: text overlap with arXiv:1906.01438

References in corpus (1)

• On Singular Points and Oscillatory Integrals

Cited by in corpus (4)

• On oscillatory integrals associated to phase functions with degenerate singular points
• On the numerical evaluation of real-time path integrals: Double exponential integration and the Maslov correction
• On asymptotic expansions of oscillatory integrals with phase functions expressed by a product of positive real power function and real analytic function in one variable
• Asymptotic expansion of oscillatory integrals with singular phases