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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
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