Fractional derivatives in complex planes

Many models are formulated in terms of fractional derivatives, such as in viscoelasticity, electrochemistry, electrode-electrolyte polarization, signal processing, diffusion processes, control processing, etc. In this paper, we first study important properties of the Caputo derivative in real line. Then we study the recently developed fractional derivative in complex plane by Ortigueira, which is very useful in signal processing. We also generalize the Caputo derivative in real line to that in complex plane then study its properties. These discussions are helpful in understanding fractional calculus and establishing fractional models in science and engineering.