Fundamental properties of fractional derivatives via pochhammer integrals

In this paper, various representations of fractional differentiation are explored, and a definition using Pochhammer contour integrals emerges as deserving special emphasis. The analyticity of Dαzpf(z) and Dαzpln z f(z) is investigated with reference to the three variables z, α, and p. The validity of the operation DβDα=Dα+α is studied. An improvement in the Leibniz rule for the fractional derivative of the product of two functions published previously is given.