  We state several equivalent noncommutative versions of the Cauchy-Riemann
equations and characterize the unbounded operators on L^2(R) which satisfy
them. These operators arise from the creation operator via a functional
calculus involving a class of entire functions, identified by Newman and
Shapiro [D. J. Newman and H. S. Shapiro, Fischer spaces of entire functions, in
Entire Functions and Related Parts of Analysis (J. Koorevaar, ed.), AMS Proc.
Symp. Pure Math. XI (1968), 360-369], which act as unbounded multiplication
operators on Bargmann-Segal space.
