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It has recently been proven that the generalised Cauchy fractional derivative(also known as the Riemann-Liouville fractional derivative) is equal to theGrunwald-Letnikov derivative. However, we observe that there are "Grunwaldnon-differentiable" functions for which the latter derivative is notconvergent, while the Riemann-Liouville derivative is. In this paper, we showthat the Riemann-Liovuille derivative can be considered a "principal value" ofthe Grunwald-Letnikov derivative, requiring specific relative rates of approachfor the limits $\Delta x \to 0$ and $N \to \infty$ (where $N$ is the upperlimit of the infinite summation) in the Grunwald derivative - i.e. instead ofvarying $\Delta x$ and $N$ independently, the two must satisfy a relation andbe varied as a single limit. We proceed to calculate this relation for severalfunctions and orders, finding that several possible such relations are possiblefor a given function, also placing requirements on the "handedness" of thefractional derivative. It is further shown that for functions with a Taylorexpansion, the relation $\Delta x=x/N$ always produces the correct principalvalue.

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