  We propose two new definitions of the exponential function on time scales.
The first definition is based on the Cayley transformation while the second one
is a natural extension of exact discretizations. Our eponential functions map
the imaginary axis into the unit circle. Therefore, it is possible to define
hyperbolic and trigonometric functions on time scales in a standard way. The
resulting functions preserve most of the qualitative properties of the
corresponding continuous functions. In particular, Pythagorean trigonometric
identities hold exactly on any time scale. Dynamic equations satisfied by
Cayley-motivated functions have a natural similarity to the corresponding
diferential equations. The exact discretization is less convenient as far as
dynamic equations and differentiation is concerned.
