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Zbl 1055.31003
Coffman, Adam; Legg, David; Pan, Yifei
A Taylor series condition for harmonic extensions. (English)
[J] Real Anal. Exch. 28, No. 1, 235-253 (2003). ISSN 0147-1937

Let $u$ be harmonic on some open ball in $\bbfR^n$ centred at the origin, and let $x= (x',x_n)$ denote a typical point of $\bbfR^n= \bbfR^{n-1}\times \bbfR$. Suppose that the Taylor series of $u(x',0)$ and $(\partial u/\partial x_n)(x',0)$ about $0'$ converge when $\vert x'\vert< r$. Then it is shown that the Taylor series of $u(x)$ converges when $\vert x'\vert+\vert x_n\vert< r$. The proof uses elementary arguments.
[Stephen J. Gardiner (Dublin)]

MSC 2000:: *31B05 Harmonic functions, etc. (higher-dimensional)
35C10 Series solutions of PDE

Keywords: harmonic function; Taylor series

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