Real and complex order integrals and derivatives operators over the set of causal functions

Abstract

The fractional integrals and fractional derivatives problem is tackled by using the operator approach. The definition domain E of operators is causal functions.Many properties of fractional integrals are given. Fractional derivatives definition is derived from fractional integrals one. Then an unified definition of fractional integrals and derivatives operator is obtained according to the sign of the real part of the order s. The study utilizes many properties of the Euler's gamma and beta functions and their extensions in R and C. Comparison with the definitions given by other authors (Liouville, Riemann,Liouville-Caputo)is done too.