  The Hadamard representation of the Green's function of a quantum field on a
curved space-time is a powerful tool for computations of renormalized
expectation values. We study the Hadamard form of the Feynman Green's function
for a massive charged complex scalar field in an arbitrary number of space-time
dimensions. Explicit expressions for the coefficients in the Hadamard
parametrix are given for two, three and four space-time dimensions. We then
develop the formalism for the Hadamard renormalization of the expectation
values of the scalar field condensate, current and stress-energy tensor. These
results will have applications in the computation of renormalized expectation
values for a charged quantum scalar field on a charged black hole space-time,
and hence in addressing issues such as the quantum stability of the inner
horizon.
