A fractional residue theorem and its applications in calculating real integrals

Abstract

As part of an ongoing effort to fractionalise complex analysis, we present a fractional version of the residue theorem, involving pseudo-residues calculated at branch points. Since fractional derivatives are non-local and fractional powers necessitate branch cuts, each pseudo-residue depends on a line segment in the complex plane rather than a single point. We demonstrate how our result can be used to evaluate many real integrals involving non-integer power functions, in a similar way to the applications of the classical residue theorem for real integrals. These calculations require deriving some elementary fractional differintegrals in a complex context, taking care of branch cuts for fully rigorous results.