Commutative Neutrix Convolution Product of Generalized Fresnel Sine Integrals and Applications

Abstract

The generalized Fresnel sine integral $S_k(x)$ and its associated functions

$S_{k+}(x)$ and $S_{k-}(x)$ are defined as locally summable functions on the real line. The generalized Fresnel sine integrals have huge applications in physics, specially in optics and electromaghetics. In many diffraction problems the generalized Fresnel integrals plays an important role. In this paper are calculated the commutative neutrix convolutions of the generalized Fresnel sine integral and its associated functions with $x^r, r=0,1,2,\dots$.