  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.
