Green function of the Poisson equation: D = 2, 3, 4

We study the Green function of the Poisson equation in two, three and four dimensions. The solution g of the equation ∇ ⃗ 2 g ( x ⃗ − x ⃗ ′ ) = δ ( D ) ( x ⃗ − x ⃗ ′ ) , where x ⃗ and x ⃗ ′ are D-dimensional position vectors, is customarily expanded into radial and angular coordinates. For the two-dimensional case (D = 2), we find a subtle interplay of the necessarily introduced scale L with the radial component of zero magnetic quantum number. For D = 3, the well-known expressions are briefly recalled; this is done in order to highlight the analogy with the four-dimensional case, where we uncover analogies of the four-dimensional spherical harmonics with the familiar three-dimensional case. Remarks on the SO(4) symmetry of the hydrogen atom complete the investigations.