Complex Grünwald–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 Grünwald–Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald–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 Grünwald–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.

Research highlights
► Formulations of Liouville, Riemann–Liouville (RL) and the Caputo (C) derivatives in the complex plane.
► Establishment of a coherent relation between those derivatives and the incremental ratio based Grunwald–Letnikov (GL).
► Deduction of regularized integral formulations
► Proposal of mixed Caputo-Grunwald-Letnikov and Riemann-Liouville-Grunwald-Letnikov derivatives.