  Working with well chosen Riemannian metrics and employing Nevanlinna's
theory, we make the thermodynamical formalism work for a wide class of
hyperbolic meromorphic functions of finite order (including in particular
exponential family, elliptic functions, cosine, tangent and the cosine--root
family and also compositions of these functions with arbitrary polynomials). In
particular, the existence of conformal (Gibbs) measures is established and then
the existence of probability invariant measures equivalent to conformal
measures is proven. As a geometric consequence of the developed thermodynamic
formalism, a version of Bowen's formula expressing the Hausdorff dimension of
the radial Julia set as the zero of the pressure function and, moreover, the
real analyticity of this dimension, is proved.
