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We establish a general version of Cauchy’s integral formula and a residue theorem for polar-analytic functions, employing the new notion of logarithmic poles. As an application, a Boas-type differentiation formula in Mellin setting and a Bernstein-type inequality for polar Mellin derivatives are deduced.

Integration of polar-analytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting