A Faster Product for Pi and a New Integral for ln(Pi/2)

TL;DR

This paper introduces a faster product formula for pi and a new integral for ln(pi/2), derived from series related to the alternating zeta function, with applications to analytic continuation and dimensional analogs.

Contribution

It presents a novel infinite product for pi and a new integral for ln(pi/2), extending previous work on related constants through series and analytic continuation.

Findings

Derived a new infinite product for pi resembling the exponential of Euler's constant

Introduced a 1-dimensional analog for ln(pi/2) based on higher-dimensional integrals

Accelerated Wallis's product using Euler's transformation

Abstract

From a global series for the alternating zeta function, we derive an infinite product for pi that resembles the product for $e^\gamma$ ($\gamma$ is Euler's constant) in math.CA/0306008. (An alternate derivation accelerates Wallis's product by Euler's transformation.) We account for the resemblance via an analytic continuation of the polylogarithm. An application is a 1-dim. analog for ln(pi/2) of the 2-dim. integrals for ln(4/pi) and $\gamma$ in math.CA/0211148.