The alpha-calculus-cum-alpha-analysis of r-th order derivative of zeta(s,alpha)

TL;DR

This paper investigates the higher order derivatives of the Hurwitz zeta function, analyzing their singularities, derivatives, primitives, and integrability, and evaluates related integrals to deepen understanding of these complex functions.

Contribution

It provides new formulae for derivatives and primitives of the Hurwitz zeta function and explores their singularities and integrability properties.

Findings

Identified the singularity structure of higher order derivatives

Derived explicit formulae for derivatives and primitives

Evaluated integrals involving derivatives of the Hurwitz zeta function

Abstract

For the higher order derivative(with respect to the first variable) of Hurwitz zeta function,we discuss as a function of the second variable,the location and the nature of its singularities and obtain the formulae for its derivative and primitive and also discuss its Riemann integrability on the unit interval and on the interval[1,infinity).We also evaluate certain integrals on the unit interval involving these functions.