Hyperbolic Extensions of Integral Formulas
 SirkkaLiisa Eriksson
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The theory of monogenic functions or regular functions is based on Euclidean metric. We consider a function theory in higher dimensions based on hyperbolic metric. The advantage of this theory is that positive and negative powers of hyper complex variables are included to the theory, which is not in the monogenic case. Hence elementary functions can be defined similarly as in classical complex analysis.
Our operator is the modified Dirac operator \(M_{k}f = Df +k \frac{Q\prime f}{x_{n}}\) , where ′ is the main involution and Qf is given by the decomposition f (x) = Pf(x) + Qf (x) e _{ n } with \(Pf (x), Qf (x)\in C\ell_{0,n}\) . The functions satisfying M _{ k } f = 0 are called khypermonogenic functions. In case k = n – 1 they are called hypermonogenic functions. The modified Dirac operator may be used to decompose the Laplace Beltrami operator \(\Delta f  \frac{k}{x_{n}}\frac{\partial f}{\partial x_{n}}\) connected to the Riemannian metric $$ds^{2}=x_{n}^{2k/(1n)} \left(dx_{0}^{2} + \ldots +dx_{n}^{2}\right)$$ .
In this paper we present a new version of the integral formulas for hypermonogenic functions and (1 – n)hypermonogenic functions valid in whole space. Since the kernels are very natural and the results are valid in the whole space the integral formulas improve the earlier results.
 Title
 Hyperbolic Extensions of Integral Formulas
 Journal

Advances in Applied Clifford Algebras
Volume 20, Issue 34 , pp 575586
 Cover Date
 20101001
 DOI
 10.1007/s0000601002112
 Print ISSN
 01887009
 Online ISSN
 16614909
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Keywords

 Primary 30G35
 Secondary 30A05
 Monogenic
 hypermonogenic
 Dirac operator
 hyperbolic metric
 Authors

 SirkkaLiisa Eriksson ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Tampere University of Technology, P.O. Box 553, FI  33101, Tampere, Finland