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Quaternionic Hyperbolic Function Theory

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Original languageEnglish
Title of host publicationTopics in Clifford Analysis
Number of pages28
ISBN (Electronic)978-3-030-23854-4
ISBN (Print)978-3-030-23853-7
Publication statusPublished - 2019
Publication typeA3 Part of a book or another research book

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


We are studying hyperbolic function theory in the skew-field of quaternions. This theory is connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric (Formula Presented) in the upper half space (Formula Presented). In the case k = 2, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function xm(m ε Z), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. We find fundamental k-hyperbolic harmonic functions depending only on the hyperbolic distance and x3. Using these functions we are able to verify a Cauchy type integral formula. Earlier these results have been verified for quaternionic functions depending only on reduced variables (x0, x1, x2). Our functions are depending on four variables.

ASJC Scopus subject areas


  • Clifford algebra, Hyperbolic Laplace operator, Hyperbolic metric, Laplace-Beltrami operator, Monogenic function, Quaternions, α-Hyperbolic harmonic, α-Hypermonogenic

Publication forum classification

Field of science, Statistics Finland