A little question on the Helmoltz Greens Function

[davon806]

Homework Statement

I was a bit confused when reading my notes.After some derivation it states that G_h = e^(ikr)G_p,where
G_h = Helmoltz Green's function
G_p = Poisson Green's function = - 1 /(4πr)

By definition (D^2)(G_p) = δ^3 (r)
(D^2 = Laplacian)

Please see the attached
(D^2)(G_p) = 0 =/= δ^3 (r),Why?

Homework Equations

The Attempt at a Solution

Included in the attached

 

[vela]

I didn't bother to check your work, but did you consider the fact that there's a singularity at r=0?