  Laplace operators perturbed by meromorphic potential on the Riemann and
separated type Klein surfaces are constructed and their indices are calculated
by two different ways. The topological expressions for the indices are obtained
from the study of spectral properties of the operators. Analytical expressions
are provided by the Heat Kernel approach in terms of the functional integrals.
As a result two formulae connecting characteristics of meromorphic (real
meromorphic) functions and topological properties of Riemann (separated type
Klein) surfaces are derived.
