Here's another approach just to illustrate the difficulty with multiple integrals that are not absolutely (i.e. $L^1$) convergent.
The integral on any finite disk centered at the origin is zero.
So it would be natural to conclude that the integral over the whole {r,z}
plane is zero.
In terms of polar coordinates:
Integrate[
((r BesselJ[0, r])/(2 (r^2 + z^2)^(1/2)) /. {z -> ρ Sin[θ], r -> ρ Cos[θ]}) ρ,
{ρ, 0, a}, {θ, 0, 2 Pi}, Assumptions -> a > 0]
(* 0 *)
Integrate[
((r BesselJ[0, r])/(2 (r^2 + z^2)^(1/2)) /. {z -> ρ Sin[θ], r -> ρ Cos[θ]}) ρ,
{θ, 0, 2 Pi}, {ρ, 0, a}, Assumptions -> a > 0]
(* Integrate[1/2 a BesselJ[1, a Cos[θ]], {θ, 0, 2 π}, Assumptions -> a > 0] *)
Quiet@Block[{a = #}, N@%] & /@ RandomReal[{0, 100}, 100] // Abs // Max
(* 3.9635*10^-14 *)
In cartesian {r, z}
coordinates:
Integrate[
(r BesselJ[0, r])/(2 (r^2 + z^2)^(1/2)),
{r, -a, a}, {z, Sqrt[a^2 - r^2], Sqrt[a^2 - r^2]}, Assumptions -> a > 0]
(* 0 *)
Integrate[
(r BesselJ[0, r])/(2 (r^2 + z^2)^(1/2)),
{r, z} ∈ Disk[{0, 0}, a], Assumptions -> a > 0]
(* 0 *)
Etc.