Series Expansion of Generalized Fresnel Integrals

2012/11/16 by Mathar, Richard J.
#33B20 (Primary) 28-04 #65D20 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1211.3963

Abstract

The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix²) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*xⁿ) for integer powers m and n, which are essentially Incomplete Gamma Functions. We generalize one step further and focus on evaluation of the integrals with kernel p(x)*exp[i*phi(x)] and polynomials p and phi. Series reversion of phi seems not to help much, but repeated partial integration leads to a first order differential equation for an auxiliary oscillating function which allows to fuse the integrals and their complementary integrals.

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