Originally Published In
Canad. J. Math. 67(2015), 241-285
We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.
Agler, Jim and McCarthy, John E., "Global Holomorphic Functions in Several Noncommuting Variables" (2015). Mathematics Faculty Publications. 17.