Lobachevsky-type Formulas via Fourier Analysis

TL;DR

This paper extends the Parseval formula to better handle Lobachevsky-type integrals, providing new identities and formulas involving cardinal sine and Bessel functions, with practical applications in analysis.

Contribution

It introduces a modified Parseval formula applicable to both periodic and non-periodic functions, enhancing the analysis of Lobachevsky-type integrals and related identities.

Findings

New Parseval formula version for practical use

Re-proved known Lobachevsky integral identities more transparently

Derived new formulas involving cardinal sine and Bessel functions

Abstract

Recently renewed interest in the Lobachevsky-type integrals and interesting identities involving the cardinal sine motivate an extension of the classical Parseval formula involving both periodic and non-periodic functions. We develop a version of the Parseval formula that is often more practical in applications and illustrate its use by extending recent results on Lobachevsky-type integrals. Some previously known, interesting identities are re-proved in a more transparent manner and new formulas for integrals involving cardinal sine and Bessel functions are given.