  Borel transformed QCD sum rules conventionally use a real valued parameter
(the Borel mass) for specifying the exponential weight over which hadronic
spectral functions are averaged. In this paper, it is shown that the Borel mass
can be generalized to have complex values and that new classes of sum rules can
be derived from the resulting averages over the spectral functions. The real
and imaginary parts of these novel sum rules turn out to have damped
oscillating kernels and potentially contain a larger amount of information on
the hadronic spectrum than the real valued QCD sum rules. As a first practical
test, we have formulated the complex Borel sum rules for the phi meson channel
and have analyzed them using the maximum entropy method, by which we can
extract the most probable spectral function from the sum rules without strong
assumptions on its functional form. As a result, it is demonstrated that,
compared to earlier studies, the complex valued sum rules allow us to extract
the spectral function with a significantly improved resolution and thus to
study more detailed structures of the hadronic spectrum than previously
possible.
