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.

Author Bio

Joakim Munkhammar is an undergraduate student at Uppsala University, he completed this paper during the spring of 2004 for Bachelor's degree of Mathematics under the supervision of his supervisor DR Andreas Stroembergsson. Joakim is currently taking astrophysics in the intent of taking a Master's in that subject.He has both Mathematics and Physics as major, with chaostheory as as favourite subject.He has been planning to take some courses in California and possibly to do some research project there, ie undergraduate thesis for some institution or company. Furthermore he is interrested in general philosophy and problemsolving. If any questions on his paper he's happy to answer them.