  We develop a theory of holomorphic functions in several noncommuting (free)
variables and thus provide a framework for the study of arbitrary n-tuples of
operators. The main topics are the following: Free holomorphic functions and
Hausdorff derivations; Cauchy, Liouville, and Schwartz type results for free
holomorphic functions; Algebras of free holomorphic functions; Free analytic
functional calculus and noncommutative Cauchy transforms; Weierstrass and
Montel theorems for free holomorphic functions; Free pluriharmonic functions
and noncommutative Poisson transforms; Hardy spaces of free holomorphic
functions.
