# A residue theorem for polar analytic functions and Mellin analogues of Boas' differentiation formula and Valiron's sampling formula

## TL;DR

This paper advances the theory of polar analytic functions by establishing a residue theorem, a version of the Cauchy integral formula, and Mellin analogues of classical formulas, with applications to differentiation, inequalities, and sampling theorems.

## Contribution

It introduces a residue theorem for polar Mellin derivatives and extends classical formulas and sampling theorems to the polar analytic function setting.

## Key findings

- Derived a residue theorem for polar Mellin derivatives.
- Extended Boas' differentiation formula to polar Mellin derivatives.
- Established an analogue of Valiron's sampling theorem for polar analytic functions.

## Abstract

In this paper, we continue the study of the polar analytic functions, a notion introduced in \cite{BBMS1} and successfully applied in Mellin analysis. Here we obtain another version of the Cauchy integral formula and a residue theorem for polar Mellin derivatives, employing the new notion of logarithmic pole. The identity theorem for polar analytic functions is also derived. As applications we obtain an analogue of Boas' differentiation formula for polar Mellin derivatives, and an extension of the classical Bernstein inequality to polar Mellin derivatives. Finally we give an analogue of the well-know Valiron sampling theorem for polar analytic functions and some its consequences.