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Khavinson, Dmitry; Lundberg, Erik; Render, Hermann
The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions
It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$ the inhomogeneous difference equation $h ( t+1,y) -h (t,y) =g ( t,y)$ has an entire harmonic solution $h$.

Keywords:reflection principle, entire harmonic function, analytic continuation
Categories:31B20, 31B05

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