# A complex analysis approach to Atangana-Baleanu fractional calculus

## TL;DR

This paper introduces a complex analysis approach to Atangana-Baleanu fractional calculus, enabling analytic continuation to complex orders and leading to more natural formulas for fractional integrals and derivatives.

## Contribution

It provides a novel complex contour integral representation of the Atangana-Baleanu derivative, extending its definition to complex orders and improving its mathematical formulation.

## Key findings

- Analytic continuation to complex orders achieved
- New formulas for fractional integrals derived
- Implications for fractional calculus theory discussed

## Abstract

The standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana-Baleanu fractional integral and for iterated Atangana-Baleanu fractional differintegrals.