Delta function at the origin in polar coordinates

Solution 1:

see http://mathworld.wolfram.com/DeltaFunction.html eqn 46. The result given there corresponds to your first equation, $\delta^{(2)}=\frac{\delta(r)}{\pi r}$. However it can be more complicated: $\delta^{(2)}=\frac{\delta(r)}{\pi r}$ is only for a Dirac Delta located at the origin.

See the pdf at https://www.google.com/#q=06_notes_2dfunctions page 18. This shows the result given in your first equation is for an Dirac Delta at the origin, but your final equation $\delta=\frac{1}{r}\delta(r-\epsilon)\delta(\theta-\theta_0)$ represents an Dirac Delta radially offset from the origin by $\epsilon$ and rotated through the angle $\theta_0$.

So your equation should work for you, maybe rewritten $\delta(r-r_0)=\frac{1}{r_0}\delta(r-r_0)\delta(\theta-\theta_0)$ where $\epsilon$ is replaced with $r_0$ and no limiting process is needed.

Consider removing 'at the origin' from your title unless you want that limitation. In that case your first equation would work.