The generalised Cauchy derivative as a principal value of the Gr\"unwald-Letnikov fractional derivative for divergent expansions

Abstract

It has recently been proven that the generalised Cauchy fractional derivative (also known as the Riemann-Liouville fractional derivative) is equal to the Grunwald-Letnikov derivative. However, we observe that there are "Grunwald non-differentiable" functions for which the latter derivative is not convergent, while the Riemann-Liouville derivative is. In this paper, we show that the Riemann-Liovuille derivative can be considered a "principal value" of the Grunwald-Letnikov derivative, requiring specific relative rates of approach for the limits $\Delta x \to 0$ and $N \to \infty$ (where $N$ is the upper limit of the infinite summation) in the Grunwald derivative - i.e. instead of varying $\Delta x$ and $N$ independently, the two must satisfy a relation and be varied as a single limit. We proceed to calculate this relation for several functions and orders, finding that several possible such relations are possible for a given function, also placing requirements on the "handedness" of the fractional derivative. It is further shown that for functions with a Taylor expansion, the relation $\Delta x=x/N$ always produces the correct principal value.