In a previous paper we considered fractional integration and differentiation of functions of a real variable. In the present paper the complex variable will be used.

We define a λth integral, or a (−λ)th differential coefficient, of f(z) along a simple curve l by

where the integration and differentiation are along l, starting from a, λ is any number, real or complex, γ is the least integer greater than or equal to zero such that R(λ) + γ > 0, R(λ) being the real part of λ; and D stands for , denoting differentiation along l. a is arbitrary and independent of z, and in the present paper is to be taken as finite.