Is there geometrical/physical meaning of the fractional integral with complex exponent?

Abstract

The geometrical/physical meaning of the temporal fractional integral with complex fractional exponent has been found and discussed. It has been shown that the imaginary part of the fractional integral is related to discrete scale invariance (DSI) phenomenon and observed only for true regular (discrete) fractals. Numerical experiments show that the imaginary part of the complex fractional exponent can be well approximated by a simple and finite combination of the leading sine/cosine log-periodical functions with period ln ξ ( ξ is a scaling parameter). In most cases analyzed, the leading Fourier components give a pair of complex conjugated exponents defining the imaginary part of the complex fractional integral. For random fractals, where invariant scaling properties are realized only in the statistical sense the imaginary part of the complex exponent is averaged and the result is expressed in the form of the conventional Riemann–Liouville integral. The conditions for realization of reind and recaps elements with complex power-law exponents have been found. Description of relaxation processes by kinetic equations containing complex fractional exponent and their possible recognition in the dielectric spectroscopy is discussed. New kinetics expressed in terms of non-integer operators with complex and real power-law exponents can be successfully applied for description of dielectric spectra of many non-crystalline solids.