This might be a duplicate but I could not find the answer here.
I want to prove the following relation for the fractional derivative $a \in \mathbb{R}$:
$$ (i\partial_x)^a = (ix)^{-a} \frac{\Gamma(a-x\partial_x)}{\Gamma(-x\partial_x)} $$
This should hold for functions that are holomorphic in $\mathbb{C}^- := \{z \in \mathbb{C}~| ~\text{Im}(z) < 0\}$ and vanish at infinity. The fractional derivative is defined by its action on the fouriertransform, i.e.
$$ (i\partial_x)^a f(x) := \int_0^{\infty} dk~k^a e^{-ikx} \widetilde{f}(k) $$ Note that the Fourier transfrom $\widetilde{f}(k)$ is $0$ for $k <0$, since $f$ is holomorphic in $\mathbb{C}^-$. So this problem reduces to show that $$ \int_0^{\infty} dk~(ikx)^a e^{-ikx} \widetilde{f}(k) = \int_0^{\infty} dk~\frac{\Gamma(a+ikx)}{\Gamma(ikx)} e^{-ikx} \widetilde{f}(k). $$



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