The integral formula of Poisson with principal value integrals and some special Gradshteyn and Ryzhik integrals

TL;DR

This paper demonstrates how Poisson's integral formula can be applied to evaluate complex integrals from Gradshteyn and Ryzhik, including principal value integrals, providing efficient solutions for special cases.

Contribution

It introduces a version of Poisson's integral formula for principal value integrals and applies it to evaluate specific integrals from classical tables.

Findings

Derived a formula for principal value integrals using Poisson's method

Successfully evaluated several Gradshteyn and Ryzhik integrals

Provided a new approach for handling singular integrals in complex analysis

Abstract

We show how the integral formula of Poisson for holomorphic functions on the right half plane can be used to quickly evaluate certain integrals from the Table of Gradshteyn and Ryzhik. In addition, we prove a version of this formula for principal value integrals and use it in several interesting cases.