Fractional Integration and Differentiation

Abstract

In this chapter we deal with “complex powers of the operator $\frac{d}{{dx}},$” a concept already found in the posthumous works of Riemann and elaborated by Marcel Riesz in the 1930s and 1940s. The relevant article [18] is lengthy, but with the help of a little distribution theory and complex analysis, all results can readily be proved.We will also deal with Riesz’s treatment of the wave operator $\Box = \partial _t^2 -\triangle_{x}$ in arbitrary dimension; thus we will obtain, among other things, a fundamental solution of $\Box.$