I am a bit stuck on solving this integral:
$$g(\omega)=\left(\frac{L}{2\pi}\right)^d\int({d^dq})\delta(\omega-cq^\alpha)$$
The range of values of $q$ is $[0,\infty).$
I have tried referencing this post $n$-dimensional integral of delta function with no success, as I do not understand the portion of rewriting the delta function in a more convenient expression.
This is what I have thus far... $$g(\omega)=\left(\frac{L}{2\pi}\right)^dVol_{d}\int({dq})q^{d-1}\delta(\omega-cq^\alpha)$$ Where $Vol_{d}=\frac{\pi^{d/2}}{(d/2)!}$
Then replace the $q^{d-1}$ in the integral with $(\frac{\omega}{c})^{1/\alpha}$
Resulting in $\left(\frac{L}{2\pi}\right)^d\frac{\pi^{d/2}}{(d/2)!}(\frac{\omega}{c})^{(d-1)/\alpha}$
My only hesitation is replacing the $q^{d-1}$ term with $(\frac{\omega}{c})^{(d-1)/\alpha}$ due to the delta function substitution.