I have been studying derivatives as operators on a function. Specifically, how we may write, for a function $f(x)$




where $D$ denotes $\frac{d}{dx}$. I've seen how successive application of an operator is seen as multiplying the operators together to give rise to a new operator. For example


can be written as $D^2f$.

What I have been pondering is this: Consider a certain operator $\Phi$, which has the property that


i.e the $\Phi$ operator acts like a sort of "square root" of the derivative operator. This would imply that

$$\Phi f = h$$


$$\Phi h = g$$

where $g(x)$ and $h(x)$ are functions and that


My question is, has such an operator been studied before? Does it have a unique name? I do not necessarily belive $\Phi$ to be unique, perhaps many unique operators may fill the role of $\Phi$. But can we determine if such an operator is unique or not?


1 Answer 1


Yes, there is a body of research about this. The correct term is Fractional Calculus and you can find more about it here: https://en.wikipedia.org/wiki/Fractional_calculus


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