Solutions to the Incomplete Toronto Function and Incomplete Lipschitz-Hankel Integrals

TL;DR

This paper derives closed-form expressions and tight bounds for the incomplete Toronto function and Lipschitz-Hankel integrals of Bessel functions, facilitating analytical studies in wireless communications.

Contribution

It introduces novel closed-form solutions and bounds for specific incomplete Bessel-related functions, applicable in wireless communication analysis.

Findings

Derived closed-form expressions for the functions.

Proposed tight upper and lower bounds.

Validated effectiveness in wireless communication applications.

Abstract

This paper provides novel analytic expressions for the incomplete Toronto function, $T_{B}(m,n,r)$, and the incomplete Lipschitz-Hankel Integrals of the modified Bessel function of the first kind, $Ie_{m,n}(a,z)$. These expressions are expressed in closed-form and are valid for the case that $n$ is an odd multiple of $1/2$, i.e. $n \pm 0.5\in\mathbb{N}$. Capitalizing on these, tight upper and lower bounds are subsequently proposed for both $T_{B}(m,n,r)$ function and $Ie_{m,n}(a,z)$ integrals. Importantly, all new representations are expressed in closed-form whilst the proposed bounds are shown to be rather tight. To this effect, they can be effectively exploited in various analytical studies related to wireless communication theory. Indicative applications include, among others, the performance evaluation of digital communications over fading channels and the information-theoretic analysis of multiple-input multiple-output systems.