Identify with .
- The 2-dimensional zonal harmonics are then given by
- Verify explicitly the properties of zonal harmonics seen in class.
- The polynomial given by is equal to for , where is the Chebyshev polynomial given by .
The Gegenbauer polynomials are given by the generating function
The polynomials , for , are given by
(Hint: Apply the operator to the generating function above, and consider the expansion of the Poisson kernel in the .)
Let be a bounded domain and f a bounded function in for some i.e. there exists some such that
Then the Newtonian potential , in , and the second derivatives of are in .