Abstract
In this work we show that it is possible to calculate the fractional integrals and derivatives of order (using the Riemann-Liouville formulation) of power functions (t-*)β with β being any real value, so long as one pays attention to the proper choice of the lower and upper limits according to the original functions domain. We, therefore, obtain valid expressions that are described in terms of function series of the type (t-*)± α+k and we also show that they are related to the famous hypergeometric functions of the Mathematical-Physics.
Highlights
The non-integer order calculus, popularly known as fractional calculus (FC) was born in 1695
We show that when one takes careful consideration on the choices for the lower and upper limits of these operations, it is possible to compute expressions for the RiemannLiouville fractional integral (RLFI) and Riemann-Liouville fractional derivative (RLFD) of any power function in terms of series that can be related to the famous hypergeometric functions [30] so important and commonly found in many problems of the Mathematical-Physics
All expressions listed on theorem for the power functions f(t)=(t−d)β are only valid when we choose the lower limit a≠d, which guarantees the convergence of the series in their respective intervals of definition
Summary
The non-integer order calculus, popularly known as fractional calculus (FC) was born in 1695. While there are many distinct formulations for a fractional differential operator [32,33], it is our hope that with this work we can, provide helpful expressions for the aforementioned calculations of RLFI and RLFD of power functions to be used on analytical or numerical related problems, and set some ground for a future discussion on the theoretical aspects of computing a fractional definite integral versus knowing its fractional primitives whenever such definitions are meaningful.
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