One of the most fundamental problems in complex
geometry is to determine when two bounded domains
in C^n are biholomorphically equivalent. Even for complete
Reinhardt domains, this fundamental problem remains unsolved
for many years. Using the Bergmann function theory,
we construct an infinite family of numerical invariants from
the Bergman functions for complete Reinhardt domains in
C^n. These infinite family of numerical invariants are actually
a complete set of invariants if the domains are pseudoconvex
with C^1 boundaries. For bounded complete Reinhardt domains
with real analytic boundaries, the complete set of numerical
invariants can be reduced dramatically although the
set is still infinite. We shall also discuss the role of the Hilbert 14th
problem in the construction of numerical biholomorphic
invariants of complete Reinhardt domains in C^n.