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Harmonic Polynomials and Dirichlet-Type Problems Sheldon Axler and Wade Ramey

Summary: 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-
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 Rn
, n > 2; the function |x|2-n
is then har-


Source: Axler, Sheldon - Dean of the College of Science and Engineering, San Francisco State University


Collections: Mathematics