A novel representation - in terms of a Laurent series - for the free e
nergy of string theory at non-zero temperature is constructed. The exa
mples of open bosonic, open supersymmetric and closed bosonic strings
are studied in detail. In all these cases the Laurent series represent
ation for the free energy is obtained explicitly. It is shown that the
Hagedorn temperature arises in this formalism as the convergence cond
ition (specifically, the radius of convergence) of the corresponding L
aurent series. Some prospects for further applications are also discus
sed. In particular, an attempt to describe string theory above the Hag
edorn temperature - via Borel analytical continuation of the Laurent s
eries representation - is provided.