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

This book, Fractional Complex Variables, are the author's lecture and research notes on the non-integer rational order derivatives 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 (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 propose and prove a theorem that shows where Strong LFCDs exist, for non-integer rational order derivatives. Applications include an index of stability for complex or real valued Fractional-Advection Dispersion Equation (FADE).