I'm trying to find out if there is a logical way to see which fractional definition is correct, so my attempt was as follows:
Suppose that we have the fact
$$\int_{0}^{\infty^{*}} \frac{\cos(ax)}{1-x^2}dx=\frac{\pi}{2} \sin(a) \ ,a>0$$
where the star above the integral is the Cauchy principle value integral.
Now Let's try to use differentiation under integral sign technique, so by differentiating both sides $p$ times with respect to a where $0<p\leq1$ using Grunwald definition we will get: $$\int_{0}^{\infty^{*}} \frac{x^p \cos(ax+\frac{\pi p}{2})}{1-x^2}dx=\frac{\pi}{2} \sin(a+\frac{\pi p}{2})$$ and we will get a correct result and this can be proved using Residue theorem or by using Laplace and Laplce inverse technique.
So my question is : Is my approach correct? , does it mean that the other fractional derivative definitions are wrong?