By adequately dividing into parts the interval of integration of the infinite integral defining the Fermi-Dirac functions ℱ a (z), new functional expansions, which determine the analytic structure of these functions in the neighbourhood of the point at infinity, are obtained. The convergence properties of these expansions are thoroughly researched in the α andz complex planes. It turns out that they converge absolutely and uniformly in wide regions of these complex planes and, from a numerical point of view, they can be used advantageoulsy to compute ℱ a (z) for intermediate and large values of |z|. This fact is especially interesting when α,z are both real and positive as a consequence of the very numerous applications that, in this case, these functions have in different branches of physics. On the other hand, the well-known asymptotic expansions for ℱ a (z) follow immediately from the obtained series expansions by neglecting in them all termsO(1/z).