  We determine the effective complex permittivity of a two-dimensional
composite, consisting of an arbitrary doubly periodic array of identical
circular cylinders in a homogeneous matrix, and whose dielectric properties are
complex-valued. Efficient formulas are provided to determine the effective
complex permittivity tensor which are in excellent agreement with numerical
calculations. We also show that in contrast to the real-valued case, the real
and imaginary parts of the effective complex-valued tensor can exhibit
non-monotonic behavior as functions of volume fraction of cylinders, and can be
either greater or less than that of the constituents.
