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.

26 pages