I do not know much about fractional calculus, except what I have read in a few short posts at MSE and https://en.wikipedia.org/wiki/Fractional_calculus.
I know that order of a derivative can be extended from rational values to real values, but all I know is what is written here.
So my question is, what is the simplest way to understand and/or define $\dfrac{d^n}{dx^n}f(x),$ for $n\in\mathbb{R}$?
For example what is $\dfrac{d^\pi}{dx^\pi}f(x)$?
Also, what about complex values? What might we say about $\dfrac{d^{s}}{dz^s}f(z)$ for $s\in\mathbb{C}$?
And what are some uses for such derivatives?
Partial answer
For $n\in\mathbb{R},$ $$D^n f(x)=D^{n-\left \lfloor{n}\right \rfloor }D^{\left\lfloor{n}\right \rfloor}f(x)=\dfrac{1}{\Gamma(1+\left \lfloor{n}\right \rfloor -n)}\displaystyle\dfrac{d}{dx}\int_0^x\dfrac{D^{\left\lfloor{n}\right \rfloor}f(t)}{(x-t)^{n-\left \lfloor{n}\right \rfloor }}\,dt,\tag{1}$$
$$D^\pi f(x)=D^{\pi-3}D^3f(x)=\dfrac{1}{\Gamma(4-\pi)}\displaystyle\dfrac{d}{dx}\int_0^x\dfrac{D^3f(t)}{(x-t)^{\pi-3}}\,dt,\tag{2}$$ and for $s,z\in\mathbb{C},$ $$D^s f(z)=?\tag{3}$$
Are at least (1) and (2) correct? Can these be `simplified?'