Harmonic Polynomials and Dirichlet-Type 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 |x| 2-n 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 Dirichlet-type 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 |x| 2-n Unless otherwise stated, we work in R n ,n> 2; the function |x| 2-n is then har- monic and nonconstant on R n \{0}. (When n = 2 we need to replace |x| 2-n with log |x|; the minor modifications needed in this case are discussed in Section 4.) Letting D j denote the partial derivative with respect to the j th coordinate vari- able, we list here some standard differentiation formulas that will be useful later: D j |x| t = tx j |x| t-2 ∆|x| t = t(t + n - 2)|x| t-2 ∆(uv)= u∆v +2∇u ·∇v + v∆u. The first two formulas are valid on R n \{0} for every real t, while the last formula holds on any open set where u and v are twice continuously differentiable (and real valued); as usual, ∆ denotes the Laplacian and ∇ denotes the gradient. 1991 Mathematics Subject Classification. 31B05. The first author was partially supported by the National Science Foundation.