Novel Expressions for the Rice $Ie{-}$Function and the Incomplete Lipschitz-Hankel Integrals

TL;DR

This paper introduces new analytic expressions and approximations for the Rice Ie-function and incomplete Lipschitz-Hankel integrals, enabling more accurate and efficient evaluations in communication system analysis.

Contribution

Novel exact series, polynomial approximations, and closed-form expressions for the Rice Ie-function and ILHIs, with tight error bounds and algebraic forms for easier application.

Findings

Derived accurate polynomial approximations valid for all parameters.

Established tight closed-form bounds for truncation errors.

Expressions facilitate analytical and numerical evaluations in communication systems.

Abstract

This paper presents novel analytic expressions for the Rice $Ie{-}$function, $Ie(k,x)$, and the incomplete Lipschitz-Hankel Integrals (ILHIs) of the modified Bessel function of the first kind, $Ie_{m,n}(a,z)$. Firstly, an exact infinite series and an accurate polynomial approximation are derived for the $Ie(k ,x)$ function which are valid for all values of $k$. Secondly, an exact closed-form expression is derived for the $Ie_{m,n}(a,z)$ integrals for the case that $n$ is an odd multiple of $1/2$ and subsequently an infinite series and a tight polynomial approximation which are valid for all values of $m$ and $n$. Analytic upper bounds are also derived for the corresponding truncation errors of the derived series'. Importantly, these bounds are expressed in closed-form and are particularly tight while they straightforwardly indicate that a remarkable accuracy is obtained by truncating each series after a small number of terms. Furthermore, the offered expressions have a convenient algebraic representation which renders them easy to handle both analytically and numerically. As a result, they can be considered as useful mathematical tools that can be efficiently utilized in applications related to the analytical performance evaluation of classical and modern digital communication systems over fading environments, among others.