**Problem 2.5** Determine the Green’s function for the annular region bounded by two concenteric spheres in

*Proof:* Suppose the annular region is and denote and , and .

Firstly

is a harmonic function in and but not necessarily 0 on .

Then is harmonic in and vanish on . Similarly construct

Then is harmonic in and vanish on . We hope that

is convergent, then is harmonic in and vanish on both boundaries. Formally we get

is well defined, because the terms of infinite sum are geometric series, it is unifomly convergent when . And very term is harmonic, thus is harmonic when . One can verify that is 0 on the .