Complex Grunwald-Letnikov, Liouville, Riemann-Liouville, and Caputo derivatives for analytic functions

The well-known Liouville, Riemann-Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grunwald-Letnikov derivative. Particularly, starting from a complex formulation of the Grunwald-Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann-Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann-Liouville and Caputo derivatives with the Grunwald-Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation. (C) 2011 Elsevier B.V. All rights reserved.