Liouville and Riemann-Liouville fractional derivatives via contour integrals

We study the fractional integral (fI) and fractional derivative (fD), attained by the analytic continuation (AC) of Liouville’s fI (LfI) and Riemann-Liouville fI (RLfI). On the AC of RLfI, we give a detailed summary of Lavoie et al’s review. The ACs of RLfI are expressed by means of contour integrals. Two of them use the Cauchy contour, and one is using the Pochhammer contour. In this case, the latter is AC of all the others for the functions treated. For the AC of LfI, one can find studies in Campos’ papers and in Nishimoto’s books, where the AC is using only the Cauchy contour. Here we present also an AC using a modified Pochhammer’s contour. In this case, we see that any of these two ACs is not the AC of the other for all the functions treated. This fact leads to difficulties, if a careful study taking care of the domains of existence of each AC is not adopted. By taking account of this fact, we resolve the difficulties which occur in Nishimoto’s formalism.