The Interplay between Fractional Calculus and Complex Analysis

The usual definitions of fractional derivatives and integrals are very well-suited for a fractional generalisation of real analysis. But the basic building blocks of complex analysis are different: although fractional derivatives of complex-valued functions and to complex orders are well known, concepts such as the Cauchy–Riemann equations and d-bar derivatives have no analogues in the standard fractional calculus. In the current work, we propose a formulation of fractional calculus which is better suited to complex analysis and to all the tools and methods associated with this field. We consider some concrete examples and various fundamental properties of this fractional version of complex analysis. Keywords: fractional derivatives, complex analysis, d-bar derivatives, Leibniz rule