Fractional Complex Variables: Strong Local Fractional Complex Derivatives (LFCDs) of Non-Integer Rational Order

Abstract

Fractional complex variables focus on the fractional or non-integer order differential calculus of a complex variable. In fractional calculus, locality can narrow down pieces of a function where there may be better behavior in order to model in an analytic sense, as well as obtain more meaningful physical and/or geometric information. That's where we introduce the concepts of Strong Local Fractional Complex Derivatives or LFCDs. Strong LFCDs can "maximize" the opportunity that the piece of the function in a localized or local enough area is "well-behaved" (enough). We prove a theorem that shows where Strong LFCDs exist. Applications include index of stability in Complex or Real Fractional Advection Dispersion Equation (FADE).