  In multicentric calculus one takes a polynomial $p$ with distinct roots as a
new variable and represents complex valued functions by $\mathbb C^d$-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 $|p(z)| \le \rho$.
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 $|p(z)| \le \rho$ 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.
