广义Cauchy导数作为发散展开式Grünwald-Letnikov分数阶导数的主值
The generalised Cauchy derivative as a principal value of the Grünwald-Letnikov fractional derivative for divergent expansions
摘要：
最近证明了广义柯西分数阶导数（也称为黎曼-刘维尔分数阶导数）等于Grunwald-Letnikov导数。然而，我们观察到有“Grunwald不可微”的功能，后者的衍生物是不收敛的，而黎曼刘维导数。在本文中，我们证明了Riemann-Liovuille导数可以被认为是Grunwald-Letnikov导数的一个“主值“，要求对极限$Δx \to 0$和$N \to \infty$有特定的相对接近率（其中$N$是无限求和的上极限）-即，代替独立地改变$Δx$和$N$，两者必须满足一个关系，并作为一个单一的限制而变化。我们继续计算这种关系的几个功能和订单，发现几个可能的这种关系是可能的一个给定的功能，也把要求的“手”的分数导数。它进一步表明，与泰勒展开的功能，关系$Δx=x/N$总是产生正确的主值。
英文摘要：
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 $Δx \to 0$ and $N \to \infty$ (where $N$ is the upper limit of the infinite summation) in the Grunwald derivative - i.e. instead of varying $Δx$ 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 $Δx=x/N$ always produces the correct principal value.
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