  A novel representation ---in terms of a Laurent series--- for the free energy
of string theory at non-zero temperature is constructed. The examples of open
bosonic, open supersymmetric and closed bosonic strings are studied in detail.
In all these cases the Laurent series representation for the free energy is
obtained explicitly.
  It is shown that the Hagedorn temperature arises in this formalism as the
convergence condition (specifically, the radius of convergence) of the
corresponding Laurent series. Some prospects for further applications are also
discussed. In particular, an attempt to describe string theory above the
Hagedorn temperature ---via Borel analytical continuation of the Laurent series
representation--- is provided.
