A coherent approach to non-integer order derivatives

The relation showing that the Grünwald-Letnikov and generalised Cauchy derivatives are equal is presented. This establishes a bridge between two different formulations and simultaneously between the classic integer order derivatives and the fractional ones. Starting from the generalised Cauchy derivative formula, new relations are obtained, namely a regularised version that makes the concept of pseudo-function appear naturally without the need for a rejection of any infinite part. From the regularised derivative, new formulations are deduced and specialised first for the real functions and afterwards for functions with Laplace transforms obtaining the definitions proposed by Liouville. With these tools suitable definitions of fractional linear systems are obtained.