Hypermonogenic Functions and their Cauchy-Type Theorems

  • Sirkka-Liisa Eriksson
  • Heinz Leutwiler
Part of the Trends in Mathematics book series (TM)


Let C n be the (universal) Clifford algebra generated by e 1,…, e n satisfying e i e j +e j ,e i =−2δ ij , i,j=1,…,n. The Dirac operator in C n is defined by \( D = \sum\nolimits_{i = 0}^n {{e_i}\frac{\partial }{{\partial {x_i}}}} \), where e0=1. The modified Dirac operator is introduced for \(k \in \mathbb{R}\) By \( {M_K}f = Df + k\frac{{Qf}}{{{x_n}}}\), where ′ is the main involution and Qf is given by the decomposition f(x)=Pf(x)+Qf(x)e n with P f (x), Q f (x)C nℒ1. A k+1-times continuously differentiable function f: Ω→C n , is called k-hypermonogenic in an open subsetΩof \( {\mathbb{R}^{n + 1}}\), if M k f (x) = 0 outside the hyperplane x n = 0. Note that 0-hypermonogenic functions are monogenic and n−1-hypermonogenic functions are hypermono-genic as defined by the authors in 15. The power function x m is hypermono-genic. The set of k-hypermonogenic functions is a right C n−1-module. We state a Cauchy type theorem for k-hypermonogenic functions. We also prove an integral formula for the P-part of an hypermogenic function.


Monogenic hypermonogenic Dirac operator hyperbolic metric 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 30A05, 30F45 


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Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Sirkka-Liisa Eriksson
    • 1
  • Heinz Leutwiler
    • 2
  1. 1.Department of MathematicsUniversity of JoensuuJoensuuFinland
  2. 2.Department of Mathematics Bismarckstrasse 1 1/2University of Erlangen-NürnbergErlangenGermany

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