  We extend the study of the Pick class, the set of complex analytic functions
taking the upper half plane into itself, to the noncommutative setting. R.
Nevanlinna showed that elements of the Pick class have certain integral
representations which reflect their asymptotic behavior at infinity. Loewner
connected the Pick class to matrix monotone functions. We generalize the
Nevanlinna representation theorems and Loewner's theorem on matrix monotone
functions to the free Pick class, the collection of functions that map tuples
of matrices with positive imaginary part into the matrices with positive
imaginary part which obey the free functional calculus.
