Caracterización de funciones fraccionarias analíticas en el sentido de Riemann-Liouville sobre estructuras hipercomplejas.

Abstract

For the context of hypercomplex numbers, a construction of calculus is presented, which allows to define an analogous theory to the one of holomorphic or analytic functions in the complex plane, stating similarities and differences. In order to achieve this task, a matrix representation of a hypercomplex number is used, allowing to use tools from standard linear algebra and to classify them. Furthermore, an introduction to fractional calculus is considered, specifically the Riemann-Liouville integral and derivative are studied and their basic properties. Taking into account these preliminaries, fractional hypercomplex analytic functions are introduced, being of special interest the particular case of generalized complex numbers. Finally, series of fractional analytic polynomial are explored, with the additional property of having one degree less when a substitute of the derivative for this context, is applied. Sufficient conditions will be given to address the theory developed for ordinary derivatives as the limit case for the theory with fractional derivatives.