  We define and study analogues of exponentials for functions on noncommutative
two-tori that depend on a choice of a complex structure. The major difference
with the commutative case is that our noncommutative exponentials can be
defined only for sufficiently small functions. We show that this phenomenon is
related to the existence of certain discriminant hypersurfaces in an irrational
rotation algebra. As an application of our methods we give a very explicit
characterization of connected components in the group of invertible elements of
this algebra.
