inverse – Special result of InverseHankelTransform

Hmmm. I'm not 100% sure because I'm not familiar with Hankel transformations. But I read that in the documents HankelTransform implicitly assumes that the input function is supported in $]0, infty[$[$[$[$, So if ring delta is not zero, r0 must be positive As I said, this is implicitly accepted; for the return value of InverseHankelTransformthis is explicitly stated by multiplication HeavisideTheta[r0], Also note that
1 / (2πr) DiracDelta[r - r0] equal 1 / (2πr0) DiracDelta[r - r0],

We can check the result InverseHankelTransform equal ring delta in the sense of distribution by integration against a symbolic test function:

Integrate[(ringDelta - InverseHankelTransform[fRingDelta, ρ, r]) φ[r],
{r, 0, ∞},
Assumptions -> r0> 0