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In this paper, we first generalize the Fresnel integrals by changing of apath for integration in the proof of the Fresnel integrals by Cauchy's integraltheorem. Next, according to oscillatory integral, we also obtain furthergeneralization of the extended Fresnel integrals. Moreover by using thisresult, we have an asymptotic expansion of an oscillatory integral with apositive real parameter, for a phase function with a degenerate critical pointexpressed by positive real power, including a moderate oscillation, and for asuitable amplitude function. This result gives a finer extension of thestationary phase method in one variable, which is known as a method for anasymptotic expansion of an oscillatory integral of a phase function with anon-degenerate critical point.

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