Double Integrals for Euler's Constant and ln(4/Pi) and an Analog of Hadjicostas's Formula

TL;DR

This paper presents new double integral analogs for Euler's constant and ln(4/Pi), along with a series expansion that characterizes ln(4/Pi) as an 'alternating Euler constant,' expanding the understanding of these constants.

Contribution

It introduces novel double integral analogs for key mathematical constants and a series expansion that interprets ln(4/Pi) as an 'alternating Euler constant,' extending classical integral formulas.

Findings

Analog double integrals for Euler's constant and ln(4/Pi)

Series expansion revealing ln(4/Pi) as an 'alternating Euler constant'

Extension of Hadjicostas's formula to new constants

Abstract

We give analogs for Euler's constant and ln(4/Pi) of the well-known double integrals for zeta(2) and zeta(3). We also give a series for ln(4/Pi) which reveals it to be an "alternating Euler constant."