Corrections, Improvements, and Comments on Some Gradshteyn and Ryzhik Integrals

TL;DR

This paper identifies errors in two integrals from Gradshteyn and Ryzhik, provides corrected and simplified formulas, extends results to complex parameters, and verifies related integrals, enhancing accuracy and applicability.

Contribution

It corrects and simplifies two specific integrals from Gradshteyn and Ryzhik, extends formulas to complex parameters, and verifies related integrals for improved accuracy.

Findings

Original formulas are incorrect under certain conditions.

New corrected and simplified formulas are derived.

Formulas are extended to complex-valued parameters.

Abstract

In this paper, we prove that two integrals from Gradshteyn and Ryzhik (2014) [1] (namely, Eqs. 3.937 1 and 3.937 2) provide incorrect results in certain conditions. We derive those conditions herein and provide the corrections required for those two formulas. We furthermore derive improved formulas for the solutions to those integrals that are less complicated, avoid the errors of the original formulas, and work under a larger range of parameter values. The improved formulas are used to verify the results of a few other related integrals from [1]; the previous results need correction in some instances, or are correct but can be extended in other instances. Lastly, we also consider the extended case of complex-valued parameters and derive the resulting formulas.