Complex Grünwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions
✍ Scribed by Manuel D. Ortigueira; Luis Rodríguez-Germá; Juan J. Trujillo
- Publisher
- Elsevier Science
- Year
- 2011
- Tongue
- English
- Weight
- 306 KB
- Volume
- 16
- Category
- Article
- ISSN
- 1007-5704
No coin nor oath required. For personal study only.
✦ Synopsis
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.