  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.
