Is Fractional Calculus Possible According to 'Fractional Calculus I'?

[Orion1]

I have located a paper claiming that it is possible to fractionally differentiate, called 'Fractional Calculus I'

Orion1 derivative integer factorial theorem:
$$\frac{d^n}{dx^n} (x^n) = n!$$

Is this paper correct? is 'Fractional Calculus' really possible?
http://nrich.maths.org/public/viewer.php?obj_id=1365&refpage=monthindex.php&part=index&nomenu=1

Fractional Integration?

Reference:
https://www.physicsforums.com/showpost.php?p=672326&postcount=1

 

[Hurkyl]

Yep! It's not nearly as nice as ordinary calculus, but you can still do it, and can apparently do cool stuff with it.

 

[lurflurf]

Fractional (and real and complex) order operators are possible and are used. Unfortunately several results that one might expect do not hold. For example
(D^n)exp(a x)=(a^n)exp(a x)
and
(D^n)cos(a x)=(a^n)cos(a x+n pi/2)
do not hold in fractional calculus.

 

[deepurple]

Yes fractional calculus is really useful tool for modeling problems in physics, biology and engineering. Actually fractional difference calculus is possible also.

 

[djeitnstine]

How does one perform a fractional derivative on a transcendental function? Although it seems quite trivial on algebraic functions.

 

[nicksauce]

The first time the idea of fractional calculus occurred to me, not knowing it was a real thing, was when I was thinking about how to calculate, in quantum mechanics, something like $<\psi |\hat{p}^n |\psi>$ where $\hat{p}=-i\hbar \frac{d}{dx}$. I was indeed quite surprised to find that fractional calculus was a real thing.

 

[matticus]

I always wondered what would happen if you substituted values other than integers (and replacing factorials with gamma functions) in cauchy's differentiation formula. Would this give the fractional derivative in the sense you guys are talking about?

 

[djeitnstine]

I always wondered what would happen if you substituted values other than integers (and replacing factorials with gamma functions) in cauchy's differentiation formula. Would this give the fractional derivative in the sense you guys are talking about?

That's exactly it sir