  We examine the validity of certain spectral integral formulas in topological
rings. We consider the sign and square-root functions in polymetric rings
containing $\frac12$. It turns out that formal analogues of classical
transformation kernels and the resolvent identity can be used to understand the
situation. In the lack of $\frac12$, the functions
$\frac12-\frac12\sgn(\frac12-z)$ and $\frac12-\sqrt{\frac14-z}$ can be
generalized, respectively.
