Origin of the fractional derivative and fractional non-Markovian continuous-time processes

A complex fractional derivative can be derived by formally extending the integer $k$ in $k\mathrm{th}$ of a function, computed via Cauchy's integral, to $\ensuremath{\alpha}$. This straightforward approach reveals fundamental problems due inherent nonanalyticity. consequence is that not uniquely defined. We explain detail anomalies (not closed paths, branch cut jumps) and try interpret their meaning physically terms entropy, friction deviations from ideal vector fields. Next, we present class non-Markovian continuous-time processes replacing standard Caputo classical Chapman-Kolmogorov governing equation Markov process. The leads replacement set exponential base functions Mittag-Leffler functions, but also creates complicated dependence structure between states. process may applied generalize Markovian SIS epidemic on contact graph more realistic setting.