On Complex Orders in Fractional Calculus: Floors, Ceilings, and Analytic Continuation

Many sources on fractional calculus include the assumption that all fractional orders are real. However, the usual definitions of Riemann–Liouville fractional integrals and derivatives can be used without modification in the case that the fractional orders are complex, and allowing complex orders creates a more rich structure in which the immense power of analytic continuation can be brought to bear to make many results on fractional derivatives trivially provable from the corresponding (easier) results on fractional integrals. This short paper summarises, for the benefit of the fractional community, the facts of using complex orders of differintegration, including the care that must be taken over defining the fractional derivative formulae precisely, and the usefulness of analytic continuation in this context.