Complex Grünwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions

Manuel Duarte Ortigueira, DEE Group Author

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)


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.
Original languageUnknown
Pages (from-to)4174-4182
JournalCommunications In Nonlinear Science And Numerical Simulation
Issue numberNA
Publication statusPublished - 1 Jan 2011

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