Bulletin of the Belgian Mathematical Society - Simon Stevin

Integral formulas for hypermonogenic functions

Sirkka-Liisa Eriksson

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Let $C\ell_{n}$ be the (universal) Clifford algebra generated by $e_{1},...,e_{n}$ satisfying $e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij}$, $i,j=1,...,n$. The Dirac operator in $C\ell_{n}$ is defined by $D=\sum_{i=0}^{n}e_{i}\frac{\partial}{\partial x_{i}}$, where $e_{0}=1$. The modified Dirac operator is introduced for $k\in\mathbb{R}$ by $M_{k}f=Df+k\frac{Q^{\prime}f}{x_{n}}$, where $^{\prime}$ is the main involution and $Qf$ is given by the decomposition $f\left( x\right) =Pf\left( x\right) +Qf\left( x\right) e_{n}$ with $Pf\left( x\right) ,Qf\left( x\right) \in C\ell_{n-1}$. A continuously differentiable function $f:\Omega\rightarrow C\ell_{n}$ is called $k$-hypermonogenic in an open subset $\Omega$ of $\mathbb{R}^{n+1}$, if $M_{k}f\left( x\right) =0$ outside the hyperplane $x_{n}=0$. Note that $0$-hypermonogenic functions are monogenic and $n-1$-hypermonogenic functions are hypermonogenic defined by the author and H. Leutwiler in [10]. The power function $x^{m}$ is hypermonogenic. We prove integral formulas of hypermogenic functions.

Article information

Bull. Belg. Math. Soc. Simon Stevin Volume 11, Number 5 (2005), 705-718.

First available in Project Euclid: 7 March 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30G35: Functions of hypercomplex variables and generalized variables
Secondary: 30A05: Monogenic properties of complex functions (including polygenic and areolar monogenic functions) 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)

Monogenic hypermonogenic Dirac operator hyperbolic metric


Eriksson, Sirkka-Liisa. Integral formulas for hypermonogenic functions. Bull. Belg. Math. Soc. Simon Stevin 11 (2005), no. 5, 705--718. https://projecteuclid.org/euclid.bbms/1110205628.

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