Estudio de funciones tipo exponencial y trigonométricas fraccionarias, y sus propiedades asociados al operador de Cauchy-Riemann fraccionario en el sentido de Riemann-Liouville, sobre estructuras hipercomplejas.

Abstract

In this thesis, Elementary type functions are characterized, specifically the exponential, sine and cosine functions. This characterization is done over a generalization of complex analysis, using fractional calculus and generalized complex numbers. With this goal in mind, fundamental algebra, complex numbers and a theory of analytic functions is revised. Then, fractional calculus in the Riemann-Liouville sense is presented, along with definitions of fractional integral and derivative, and relevant results for the scope of this work. After this, generalized complex numbers are introduced, with its algebraic properties, trigonometry and theory of analytic functions, comparing with known results of complex analysis. This theory of generalized complex analytic functions can be extended in two dimensions, using fractional calculus, to define a theory of fractional generalized analytic functions which allows the characterization of the fractional exponential and fractional trigonometric type functions over generalized complex numbers.