On the use of the variable change w=exp(u) to establish novel integral representations of the Riemann zeta(s,a) -function, incomplete gamma- function, confluent hypergeometric Phi-function and beta function

TL;DR

This paper introduces new integral representations for special functions like the incomplete gamma, hypergeometric, confluent hypergeometric, and beta functions using a variable change, providing insights into the Riemann zeta function.

Contribution

The paper presents novel integral representations of key special functions via the variable change w=exp(u), offering new analytical tools and a pedagogical proof for the Riemann zeta function's functional relation.

Findings

New integral representations for special functions.

A pedagogical proof of the Riemann zeta function's functional relation.

Derivation of Hurwitz's representation of the zeta function.

Abstract

The variable change w=exp(u) is applied to establish novel integral representations of the incomplete gamma-function, hypergeometric F-function,confluent hypergeometric /Phi-function and beta-function, and to analyze these functionsas as well as the Riemann /zeta(s,a)-function. In particular, using these representations we give a pedagogically instructive proof of the well known approximate functional relation for the Riemann /zeta-function and derive Hurwitz representation of the /zeta(s,a)-function.