Series Expansion of Generalized Fresnel Integrals

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.