# Fractional Derivative Implications/Meaning?

I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of the function's slope, and the second derivative is representative of its concavity, is there any qualitative relationship between a 1/2 derivative and its original function? Or a 3/2 derivative with its respective function?

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–  Amzoti Feb 22 at 0:08
Thanks @Amzoti, for the resource. This may reveal my ignorance, but I wasn't really aware of a subfield "fractional-calculus (like I said, this may simply reveal a certain degree of new-bie ignorance on my part!) –  amWhy Feb 22 at 0:13
@AmWhy: I am also ignorant on the subject - so found that nice paper to try and put context on it - and it sounds like it is a relatively new area - since the paper is from 2007. Regards –  Amzoti Feb 22 at 0:20
+ And the question is a good question! Not to mention my lack of familiarity with fractional-calculus, the question of understanding what that means, intuitively, is a question very worth the asking! –  amWhy Feb 22 at 0:23
The topic shows up in Laplace transform as a curiosity. $L[(-t)^n f(t)]= F^{(n)}(s)$. Now presumably you can take $n=1/3$ for example. –  Maesumi Feb 22 at 4:53
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(Non integer) fractional derivatives are non-local so the 1/2 derivative can not have a local meaning like tangent or curvature but would have to take into account the properties of the curve over a large extent (boundary conditions). See http://en.wikipedia.org/wiki/Fractional_calculus#Nature_of_the_fractional_derivative

In addition some of the applications of fractional derivations (http://en.wikipedia.org/wiki/Fractional_calculus#Applications) show the physical meaning is non-local.

Why is the fractional derivative non-local? Integrating a function is not unique or local, because it depends the values of the function over the entire range of integration. The generalization to fractional derivatives unifies differential and integral operators into one Differintegral operator. Whole derivatives are both unique and local. The apparent paradox of fractional derivatives being non-local is actually the natural case, just as integration is non-local.

There is a neat demonstration of calculating the fraction derivative from first principles as a limit in this article http://mathpages.com/home/kmath616/kmath616.htm and it shows how in the whole derivative case the extra terms zero out, but in the fractional case they do not.

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Very interesting, but why must the fractional derivative be non-local? The Wikipedia article doesn't seem to give a justification. –  Jonathan Feb 26 at 22:58
I added to my answer to say why it is non-local –  Michael Smith Feb 27 at 3:42
Just because it's non-local doesn't mean it doesn't have a geometric interpretation. Also, is there a concept of $\frac{d^b}{da^b} \frac{d^a}{dx^a}$? –  ChickenGod May 23 at 4:37
You are right it can have a geometric interpretation just as the single integral can be interpreted as the area under the curve. And it is not a local geometric property as the tangent is. I am having trouble reading the small print in your question and if you are asking can you have arbitrary real numbers in the power of the derivative operator then I believe the answer is yes. Actually I have read you can even put complex numbers in there! –  Michael Smith May 23 at 12:30