A Taylor series condition for harmonic extension

Adam Coffman, David Legg, and Yifei Pan

article abstract

For a harmonic function on an open subset of real n-space, we propose
a condition on the Taylor expansion that implies harmonic extension to
a larger set, by a result on the size of the domain of convergence of
its Taylor series.  The result in the n=2 case is due to M. Bocher
(1909), and the generalization to n>2 is given a mostly elementary
proof, using basic facts about multivariable power series.