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Hyperbolic Function Theory in the Skew-Field of Quaternions

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Original languageEnglish
Article number97
JournalAdvances in Applied Clifford Algebras
Issue number5
Publication statusPublished - 1 Nov 2019
Publication typeA1 Journal article-refereed


We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called α-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric dsα2=dx02+dx12+dx22+dx32x3αin the upper half space R+4={(x0,x1,x2,x3)∈R4:x3>0}. If α= 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∈Z), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental α-hyperbolic harmonic functions, depending only on the hyperbolic distance and x3, we verify a Cauchy type integral formula for conjugate gradient of α-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued α-hypermonogenic in the Clifford algebra Cℓ0,3.

ASJC Scopus subject areas


  • Clifford algebra, Hyperbolic Laplace operator, Hyperbolic metric, Laplace–Beltrami operator, Monogenic function, Quaternions, α-hyperbolic harmonic, α-hypermonogenic

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