International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 48391, 12 pages

Riesz potential operators and inverses via fractional centred derivatives

Manuel Duarte Ortigueira1,2

1UNINOVA, Campus da FCT da UNL, Quinta da Torre, Monte de Caparica 2825–114, Portugal
2Departamento de Engenharia Electrotécnica, Faculdade de Ciências e Tecnologia, Universidade de Lisboa, Monte de Caparica, Caparica 2829–516, Portugal

Received 2 January 2006; Revised 4 May 2006; Accepted 7 May 2006

Copyright © 2006 Manuel Duarte Ortigueira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well-known Grünwald-Letnikov derivatives.