Rational functions as new variables

In multicentric calculus, one takes a polynomial p with distinct roots as a new variable and represents complex valued functions by C^d-valued functions, where d is the degree of p. An application is e.g. the possibility to represent a piecewise constant holomorphic function as a convergent power series, simultaneously in all components of | p(z) | ≤ ρ. In this paper, we study the necessary modifications needed, if we take a rational function r= p/ q as the new variable instead. This allows to consider functions defined in neighborhoods of any compact set as opposed to the polynomial case where the domains | p(z) | ≤ ρ are always polynomially convex. Two applications are formulated. One giving a convergent power series expression for Sylvester equations AX- XB= C in the general case of A, B being bounded operators in Banach spaces with distinct spectra. The other application formulates a K-spectral result for bounded operators in Hilbert spaces.