We continue here our work on fractional integration and differentiation of functions of a real variable which we began in a previous paper. All quantities in the present paper are real.

We define a λth integral, or a (−λ)th differential coefficient, of f(x) over an interval (a, x)by

where D stands for and γ is the least integer greater than or equal to zero such that λ + γ>0, f(x) is bounded on the path of integration, and is continuous on this path except possibly for a finite number of ordinary discontinuities.