# "Square root" of a derivative operator?

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

$$\frac{df}{dx}$$

as

$$Df$$

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

$$\frac{d^2f}{dx^2}$$

can be written as $$D^2f$$.

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

$$\Phi^2f=Df$$

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

$$\Phi f = h$$

and

$$\Phi h = g$$

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

$$\frac{df}{dx}=g$$

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?

• You mean this? en.wikipedia.org/wiki/Fractional_calculus Jul 12, 2019 at 10:46
• @MattiP. wow thank you for that link. Can you please somehow convert it into an answer so I can accept it? Jul 12, 2019 at 10:47
• Jul 12, 2019 at 11:08
• There is a nice channel on youtube by Dr. Peyam. He has several videos related to the fractional calculus and the reasoning behind defining some of the concepts. Here is a playlist for reference Jul 12, 2019 at 14:08