Two Neumann Series Expansions for the Sine and Cosine Integrals

2016/06/30 by Chance Sanford, Sanford, Chance
Mathematics · #Advanced Mathematical Identities #Mathematical functions and polynomials #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.1607.05106

Abstract

In this work, series expansions in terms of Bessel functions of the first kind are given for the sine and cosine integrals. These representations differ from many of the known Neumann-type series expansions for the sine and cosine integrals, which contain non-integer order or quadratic Bessel function terms. In addition, using the theory of Euler sums we are able to obtain some closed form evaluations of integrals involving Bessel functions and the sine and cosine integrals.

Related

• Integral representation for Bessel's functions of the first kind and\n Neumann series
• Integral representations and asymptotic expansions for the second type Neumann series of Bessel functions of the first kind
• Some applications of the sine and cosine integrals
• Infinite series representations for Bessel functions of the first kind of integer order
• Definite integrals involving Bessel functions expressed as a series of special functions