In the sense of distributions, the Green function of the Helmholtz equation $$(\nabla^2+k^2)G(r)=\delta(r)$$ is an spherical wave $$G(r)=-\frac{1}{4\pi}\frac{e^{ikr}}{r}\,.$$
How can I prove that a distribution like $$G(x,y,z)=e^{ikz}\frac{e^{(ik/2z)[x^2+y^2]}}{z}\,,$$
which is is a wave with parabolic phase, does not equals a Dirac delta $\delta$ but correspond to a different distribution.