Abstract
Recently, Katugampola (Appl. Math. Comput. 218:860-865, 2011) studied a special case of the Erdélyi-Kober generalized fractional derivative. This special case generalized the well-known Riemann-Liouville and the Hadamard fractional integrals into a single form. Katugampola denoted this special case by the operator D x α 0 ρ . Some properties and examples for this fractional derivative operator was given. In this paper, we present some additional properties for this operator defined, this time, in the complex plane. In particular, we express this fractional derivative operator in terms of the classical Riemann-Liouville fractional derivative operator. A generalized Leibniz rule is obtained.MSC:26A33, 33C45.
Highlights
One of the most frequently encountered operators of fractional derivatives, that is, calculus of integrals and derivatives of an arbitrary real or complex order is provided by the Riemann-Liouville operator [, ] denoted Dαz and defined by
The second representation is obtained by making use of a new transformation formula for the fractional derivative recently published by Tremblay et al [ ]
This is done by using the first representation ( ) and the generalized Leibniz rule for fractional derivatives obtained by
Summary
Using a single-loop contour of integration, we can obtain a less restrictive definition for the fractional derivative operator Dαz which holds for Re(p) > – and α not a negative integer This representation has been widely used in many interesting papers [ – ]. The less restrictive representation for the fractional derivative operator in the complex plane is the one introduced by Lavoie, Osler and Tremblay [ ] in based on Pochhammer’s contour of integration [ – ]. This representation is valid when α is not a negative integer and p is not an integer.
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