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

Accurate relationships between fractals and fractional integrals: New approaches and evaluations

“…These new results we consider as a natural generalization of the previous and approximate results achieved in papers [11,10].…”

“…In order to show it let us reproduce some mathematical expressions that will be helpful for further manipulations and understanding the problem posed. As it has been shown in [10] the Laplace image of the kernel of the Cantor set is described by expression…”

“…In paper [10] the relationship between fractional integral with complex power-law exponent and Cantor set has been established. But this relationship was approximate and obtained in one-mode approximation and it would be desirable to establish the exact relationship between 1D fractals and fractional integrals in time-domain.…”

“…The log-periodic function π ν (ln z ± ln ξ) = π ν (ln z) with period ln(ξ) that figures in 1 can be decomposed to the infinite Fourier series [9,10] …”

“…One of the basic problems that did not accurately solved yet in the fractional calculus community is the finding of the justified and accurate relationships between the smoothed functions averaged over fractal objects and fractional operators. This problem was solved partly for the time-dependent functions averaged over Cantor sets in monograph [9] and paper [10], where the influence of unknown log-periodic function (leading finally to the understanding of the meaning of the fractional integral with the complex-conjugated power-law exponents) was taken into account. Possible generalizations helping to understand the role of a spatial fractional integral as a mathematical operator replacing the operation of averaging of the smoothed functions over fractal objects were considered in monograph [9] as well.…”