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would be possible to define the fractional derivative (and integral) as

$$ D^{a}f(x)=F.P\frac{1}{\Gamma(-a)}\int_{c}^{x}dt \frac{f(x)-f(t)}{(x-t)^{1+a}}$$

here c ,a are real constant (a can be negative or positive) is this a formal good definition for the fractional derivative ?? :D , here F.P means the finite part of the integral

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What you have in there is equivalent to $$\frac{f(x)}{\Gamma(1-a)(x-c)^a}-\color{red}{\frac1{\Gamma(-a)}\int_c^x \frac{f(t)}{(x-t)^{a+1}} \mathrm dt}$$ where the latter term (highlighted in red) is the usual Riemann-Liouville differintegral. – J. M. Jul 13 '12 at 10:39

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