A fractional residue theorem and its applications in calculating real integrals

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.