  A definition of hyperbolic meromorphic functions is given and then we discuss
the dynamical behavior and the thermodynamic formalism of hyperbolic functions
on the Julia set. We prove the important expanding properties for hyperbolic
functions on the complex plane and with respect to the euclidean metric. We
establish the Bowen formula for hyperbolic functions on the complex plane, that
is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia
set and furthermore, we prove that all the results in the Walters' theory hold
for hyperbolic functions on the Riemann sphere.
