I was doing some integrals in arbitrary dimensions, and I know the answer I am supposed to get, but I cant seem to get there.
The integral is
$$\int d^D k e^{i\vec{k} \cdot \vec{r}} \frac{1}{k^2+m^2}$$
I know that I should tranform the dot product as $kr \cos( \theta)= kru$, but when I do this, I get $$ \int d k \hspace{0.25 cm }k^{D-1} \int du \hspace{0.25cm} e^{i k r u} \frac{1}{k^2 +m^2}= \frac{1}{i r} \int dk \hspace{0.25 cm} k^{D-2} \frac{\sin(kr)}{k^2 +m^2}$$ I always just end up with a 1/r dependence when doing the contour integration in the upper half plane instead of the $1/r^{D-1}$ that would be correct.
Where am I making the mistake?