Fractionalisation of complex d-bar derivatives

Abstract

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

AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 30A99

Acknowledgements

The first author would like to thank Maria Christina van der Weele for useful recommendations to the literature. Additionally, the authors would like to thank the anonymous reviewers for their helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).