  Series of extended Epstein type provide examples of non-trivial zeta
functions with important physical applications. The regular part of their
analytic continuation is seen to be a convergent or an asymptotic series. Their
singularity structure is completely determined in terms of the Wodzicki residue
in its generalized form, which is proven to yield the residua of all the poles
of the zeta function, and not just that of the rightmost pole (obtainable from
the Dixmier trace). The calculation is a very down-to-earth application of
these powerful functional analytical methods in physics.
