Resolving the arbitrariness of the Euler summation through the deformed operator logarithm

Abstract

We investigate operator logarithms beyond the classical domain of validity for the Mercator series, specifically addressing elements of a unital Banach algebra whose spectra lie partially outside the standard disk of convergence. For the case of a unitary involution U, (U² = I), traditional regularization via Euler summation assigns finite values to divergent series, but relies on an ad hoc path that lacks intrinsic justification. To resolve this arbitrariness, we present a framework where regularization arises naturally from a one-parameter deformation of the logarithm defined via the holomorphic functional calculus of a unital Banach algebra. This deformed operator logarithm induces a convergent power series whose analytic continuation reproduces the Euler-summed Mercator series in a singular limit of the deformation parameter. Our findings suggest a broader paradigm in which operator expansions are understood as deformed analytic continuations, effectively unifying classical summation techniques with the holomorphic functional calculus.