On an approach to complex analysis via differintegrations

Abstract

The ordinary concept of derivative (primitive) of order n has been extended from (v = n) (v = —n) a positive (negative) integer n IN to fractional or complex v differintegration operators applied to real [1, 2], complex [3, 4] and generalized [5, 6] functions. We recall (sec. 1) the definition of Liouville (Riemann) differintegration Dv (dv ) of an analytic function (function with one branch-point), and use the corresponding rules of derivation with complex order to solve (sec. 2) a class of linear fractional integro-differential equations with constant (power) coefficients: this leads to some results on the number of roots of a class of relations between Gamma functions (sec. 7). We proceed to compare the rules of differintegration in the Liouville and Riemann systems, and to introduce (sec. 3) a branch-point derivative δ P–v which annihilates the Liouville derivative Dv for all non-integer v ∊ C — Z and integer p ∊ Z. We review (sec. 4) the generalization of Taylor's, Laurent's, Lagrange's and Teixeira'...