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A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory

  • Sirkka-Liisa ErikssonEmail author
  • Heikki Orelma
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this paper we study a mean-value property for solutions of the Laplace-Beltrami equation
$$x^{2}_{n} \Delta h - (n-1) x_n \frac{\partial h}{\partial x_n} = 0$$
(Equation 1 )
with respect to the volume and the surface integral on the Poincaré upper-half space \(\mathbb{R}^{n+1}_{+} = \{(x_0,...,x_n)\ \in \mathbb{R}^{n+1} : x_n > 0\}\) with the Riemannian metric \(g = \frac{dx^{2}_{0} + dx^{2}_{1} +...+ dx^{2}_{n}}{x^{2}_{n}}\). We also compute the Cauchy type kernels in terms of the hyperbolic metric.

Keywords

Laplace-Beltrami operator mean value theorem hypermonogenic function 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsTampere University of TechnologyTampereFinland

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