Garabedian’s Way for the Polyharmonic Green–Almansi Function

An explicit formula for the polyharmonic Green function of any order for the unit disc of the complex plane based on the work of E. Almansi is widely known through I.N. Vekua’s 1967 book. A simple, however, not easy way to prove the formula is by verification of the defining properties. P.R. Garabedian has suggested a constructive way for this function in his 1964 pde book but only has applied it for the biharmonic case. His construction is not smoothly working for higher order as is explained here in the triharmonic case. The general case will demand very concentrated calculations. The procedure should similarly work also for the half plane.