ON THE ‘PROPER’ PRIMITIVES IN FRACTIONAL CALCULUS

For the definition of fractional order derivatives it seems not to be sufficient, that these derivatives coincide with the classical derivatives in the case their orders become integers, and that the classical first order derivative of the fractional derivative of order α equals the fractional derivative of order α+1. We could further ask for the existence and uniqueness of ‘proper’ primitives for indefinite integrals of fractional order. The possibilities to define such proper primitives are discussed. A transformation will be presented allowing the introduction of generalised complex derivatives to any arbitrary complex order yielding directly such proper primitives.