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.