Two Neumann Series Expansions for the Sine and Cosine Integrals

TL;DR

This paper introduces new Neumann series expansions for sine and cosine integrals using Bessel functions, offering alternative representations and closed-form evaluations of related integrals.

Contribution

It presents novel Neumann series expansions involving integer-order Bessel functions and derives closed-form integral evaluations using Euler sums.

Findings

New series expansions for sine and cosine integrals

Alternative Bessel function representations differ from existing series

Closed-form evaluations of integrals involving Bessel and sine/cosine integrals

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.