Composition of Fractional Integral and Derivative Operators: Summarised in Tables

This paper compiles a complete, detailed list of composition properties for Riemann–Liouville fractional differintegrals, in all possible cases for orders anywhere in the complex plane, with the results presented clearly in a table for easy visual consumption. The principle of analytic continuation for fractional differintegrals, generally underappreciated in the literature, helps to make the results more concise and natural. These results for Riemann–Liouville operators are well known classically, so here are also presented, for the first time, the corresponding results for the general transmuted operators, which include as special cases all fractional differintegrals taken with respect to monotonic functions, of both left‐sided and right‐sided types. For verifiability and self‐containedness, full proofs are included, even for the classical results, as an appendix.