In this paper we give some background theory on the concept of fractional calculus, in particular the Riemann-Liouville operators. We then investigate the Taylor-Riemann series using Osler's theorem and obtain certain double infinite series expansions of some elementary functions. In the process of this we give a proof of the convergence of an alternative form of Heaviside's series. A Semi-Taylor series is introduced as the special case of the Taylor-Riemann series when \alpha=1/2, and some of its relations to special functions are obtained via certain generating functions arising in complex fractional calculus.