zbMATH — the first resource for mathematics

A Taylor series condition for harmonic extensions. (English) Zbl 1055.31003
Let $$u$$ be harmonic on some open ball in $$\mathbb{R}^n$$ centred at the origin, and let $$x= (x',x_n)$$ denote a typical point of $$\mathbb{R}^n= \mathbb{R}^{n-1}\times \mathbb{R}$$. Suppose that the Taylor series of $$u(x',0)$$ and $$(\partial u/\partial x_n)(x',0)$$ about $$0'$$ converge when $$| x'|< r$$. Then it is shown that the Taylor series of $$u(x)$$ converges when $$| x'|+| x_n|< r$$. The proof uses elementary arguments.
MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 35C10 Series solutions to PDEs
Keywords:
harmonic function; Taylor series
Full Text: