In[]:=
(Sqrt[r0^2]DiracDelta[-Sqrt[r0^2-rho^2]+z])/Sqrt[r0^2-rho^2]-(Sqrt[r0^2]DiracDelta[Sqrt[r0^2-rho^2]+z])/Sqrt[r0^2-rho^2]//FullSimplify
Out[]=
2
r0
2
r0
2
rho
2
r0
2
rho
2
r0
2
r0
2
rho
2
r0
2
rho
In[]:=
FactorTogetherDiracDelta--+z--DiracDelta-+z-
2
r0
2
r0
2
rho
2
r0
2
rho
2
r0
2
r0
2
rho
2
r0
2
rho
Out[]=
1
2
r0
2
rho
2
r0
2
r0
2
rho
2
r0
2
rho