fractional derivative of a heaviside function given the function
$$ f(x)= \frac{H(x+1)}{\sqrt{x+1}} $$
how can i evaluate the fractional derivative 
$$ \frac{d^{1/2}}{dx^{1/2}}f(x) $$
if i use the standar definition for powers of 'x' i get a coefficient $ \frac{\Gamma(1/2)}{\Gamma(0)} $ so apparently the derivative would be 0 but i think it should be something about $ \delta (x+1) $ since the half derivative applied two times is just the ordinary derivative so what is the answer ??

Hint: Consider the Fourier transform of that function.  The $p$th derivative is the inverse F.T. of $(i 2 \pi x)^p$ times the F.T. of the function. 
EDIT
More detail: for $f(x) = \frac{\mathrm{H}(x+1)}{\sqrt{x+1}} $, the F.T. is very simple:
$$\hat{f}(v) = |v|^{-\frac{1}{2}} \exp{(i 2 \pi v)} $$
so that the $p$th derivative $f^{(p)}(x)$ is
$$ f^{(p)}(x) = (i 2 \pi)^p  \int_{-\infty}^{\infty} dv \: v^p |v|^{-\frac{1}{2}} \exp{[i 2 \pi (x+1) v]} $$
You can do this last integral analytically (this is where I run out of time), and this is your fractional derivative of your function.  I am aware that $\Im{f^{(p)}(x)}$ should be 0, but it is not obvious until the integral is worked out that this will be so.
