Operator of fractional derivative in the complex plane

The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of the general derivative operator acting on the functions analytic on a curve in the complex plane is deduced and the correspondence with some well known approaches is shown. In particular. it is shown how the uniqueness of the operation depends on the derivative order type (integer, rational, irrational, complex) and the number of poles of the considered function in the complex plane.