fractional derivatives

**Random Variable** · August 3rd 2009, 08:13 PM

My knowledge of fractional calculus is very limited. But from what I understand, the general defintion of the nth derivative of a function f(x) (a defintion that is inexplicably not given in most calculus textbooks)

$D^{n}f(x) = \lim_{\Delta x \to 0} \frac {1}{(\Delta x)^{n}} \sum^{n}_{j=0} \binom{n}{j} f(x-j \Delta x)$

can be extended to derivatives of non-integer order by use of the generalized binomial theorem

$D^{\nu}f(x) = \lim_{\Delta x \to 0} \frac {1}{(\Delta x)^{\nu}} \sum^{\infty}_{j=0} \frac{\Gamma(\nu+1)}{j! \Gamma(\nu -j+1)} f(x-j \Delta x)$

Even though this seems like a logical extension to make, and expected results occurs (e.g., $\frac{d^{1/2}}{dx^{1/2}} \Big(\frac{d^{1/2}}{dx^{1/2}}f(x)\Big) = \frac {d}{dx}f(x)$ ) is there a geometric interpretation of fractional derivatives?

Also, is it correct to say that integration is then just a specific type of differentiation?

$\frac{d^{-1}}{dx^{-1}} \Big(\frac{d^{1}}{dx^{1}}f(x)\Big) = \frac{d^{0}}{dx^{0}}f(x) = f(x)?$

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