Series Expansion of Generalized Fresnel Integrals
Classical Analysis and ODEs
2012-12-05 v2
Abstract
The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) 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.
Keywords
Cite
@article{arxiv.1211.3963,
title = {Series Expansion of Generalized Fresnel Integrals},
author = {Richard J. Mathar},
journal= {arXiv preprint arXiv:1211.3963},
year = {2012}
}
Comments
Remark 3 added. Corrected denominator in eq. (3.17) and enumeration of tables. Expanded references