Operational Mathematics of Differentiation and Integration: Continuous Extension of Operation Count, Analytic Continuation, and a Unified Theory

This paper systematically extends the core methodology of Operational Mathematics—the extension of the repetition count of fundamental operations from natural numbers to integers, rationals, reals, and complex numbers—to the linear operators of differentiation D and integration I. We establish a complete axiomatic system for the families {Dα}α∈C and {Iα}α∈C, rigorously define integer-, fractional-, real-, and complex-order differential and integral operators, and employ the Laplace and Mellin transforms as linear analogues of the Schröder and Abel functional equations. Using convolution semigroups and the Hille Yosida theory of infinitesimal generators, we prove the existence of iterations at every level and establish uniqueness theorems under natural analyticity conditions. We thoroughly analyze the singularity structure of complex-order operators, proving that for functions in the Schwartz class, the map α → Dαf is meromorphic with simple poles at the negative integers (under general conditions), and we construct the associated Riemann surface. Furthermore, we rigorously prove the existence of a natural boundary along the negative real axis for a dense class of compactly supported smooth functions. The differential and integral equations of classical and fractional calculus are reformulated as evolution equations driven by the infinitesimal generator lnD, unifying discrete iteration and continuous flow under the umbrella of the operation count. We establish a categorical equivalence between the differential–integral operator semigroup and the hyperoperation iteration semigroup, thereby extending the number–operation duality to the linear realm.