 
Summary: Harmonic Polynomials and DirichletType Problems
Sheldon Axler and Wade Ramey
30 May 1995
Abstract. We take a new approach to harmonic polynomials via differ
entiation.
Surprisingly powerful results about harmonic functions can be obtained simply
by differentiating the function jxj 2\Gamman and observing the patterns that emerge. This
is one of our main themes and is the route we take to Theorem 1.7, which leads
to a new proof of a harmonic decomposition theorem for homogeneous polynomials
(Corollary 1.8) and a new proof of the identity in Corollary 1.10. We then discuss
a fast algorithm for computing the Poisson integral of any polynomial. (Note: The
algorithm involves differentiation, but no integration.) We show how this algorithm
can be used for many other Dirichlettype problems with polynomial data. Finally,
we show how Lemma 1.4 leads to the identity in (3.2), yielding a new and simple
proof that the Kelvin transform preserves harmonic functions.
1. Derivatives of jxj
2\Gamman
Unless otherwise stated, we work in R n ; n ? 2; the function jxj 2\Gamman is then har
monic and nonconstant on R n n f0g. (When n = 2 we need to replace jxj 2\Gamman with
log jxj; the minor modifications needed in this case are discussed in Section 4.)
