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More on Modified Spherical Harmonics

  • Heinz LeutwilerEmail author
Article
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Part of the following topical collections:
  1. T.C. FTHD 2018

Abstract

A modification of the classical theory of spherical harmonics is presented. The space \({\mathbb {R}}^d = \{(x_1,\ldots ,x_d)\}\) is replaced by the upper half space \({{\mathbb {R}}}_{+}^{d}=\left\{ (x_1,\ldots ,x_d), x_d > 0 \right\} \), and the unit sphere \(S^{d-1}\) in \({\mathbb {R}}^d\) by the unit half sphere \(S_{+}^{d-1}=\left\{ (x_1,\ldots ,x_d): x_1^2 + \cdots + x_d^2 =1, x_d > 0 \right\} \). Instead of the Laplace equation \(\Delta h = 0\) we shall consider the Weinstein equation \(x_d\Delta u + k \frac{\partial u }{\partial x_d}= 0\), for \(k \in {\mathbb {N}}\). The Euclidean scalar product for functions on \(S^{d-1}\) will be replaced by a non-Euclidean one for functions on \(S_{+}^{d-1}\). It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In case \(k=d-2\) the modified theory has already been treated by the author.

Keywords

Spherical harmonics Generalized axially symmetric potentials Modified spherical harmonics 

Mathematics Subject Classification

Primary 30G35 Secondary 33A45 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFriedrich-Alexander-University Erlangen-NurembergErlangenGermany

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