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Fractional derivatives in complex planes. (English) Zbl 1173.26305
Some properties regarding the Caputo derivative defined on real lines are studied. These properties include the expression of the Caputo derivative operator in analytic function space, and homogeneous property. Then some properties such as consistency, compositions of the Ortigueira derivative defined in the complex plane are obtained. Further the Caputo derivative on the real line is generalised to that on the complex plane. The characters of the Caputo derivative on the complex plane which includes non-consistency with classical derivatives, composition properties and its Laplace and Fourier transforms are also discussed. Finally the corresponding Riemann-Liouville derivative defined in complex palnes is considered.

MSC:
26A33 Fractional derivatives and integrals
46F12 Integral transforms in distribution spaces
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