Cauchy's integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order

The modified Riemann-Liouville fractional derivative applies to functions which are fractional differentiable but not differentiable, in such a manner that they cannot be analyzed by means of the Djrbashian fractional derivative. It provides a fractional Taylor's series for functions which are infinitely fractional differentiable, and this result suggests introducing a definition of analytic functions of fractional order. Cauchy's conditions for fractional differentiability in the complex plane and Cauchy's integral formula are derived for these kinds of functions. (C) 2010 Elsevier Ltd. All rights reserved.