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

Westudy theGreen functionof thePoisson equation in two, three and four dimensions. The solution g of the equation g x x x x D 2 d  ¢ = ¢      ( ) ( ) ( ) , where x and x ¢  areD-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 scaleLwith the radial component of zeromagnetic quantumnumber. ForD=3, thewell-known expressions are briefly recalled; this is done in order to highlight the analogywith the four-dimensional case,whereweuncover analogies of the fourdimensional spherical harmonicswith the familiar three-dimensional case. Remarks on theSO(4) symmetry of the hydrogen atomcomplete the investigations.