Computational Methods and Function Theory 5 (2005), No. 2, 395--408
Copyright Heldermann Verlag 2005
Sirkka-Liisa Eriksson
sirkka-liisa.eriksson@tut.fi , Institute of Mathematics, Tampere University of Technology, P.O.Box 53, 33101 Tampere, Finland.
Jani Hirvonen
jani.hirvonen@tut.fi , Institute of Mathematics, Tampere University of Technology, P.O.Box 53, 33101 Tampere, Finland.
Identify $\mathbb{C}^3$ with the set of reduced biquaternions of the form ${z=z_0+z_1e_1+z_2e_2}$, where $z_i\in\mathbb{C}$. For $\Omega\subset\mathbb{C}^3$, let $w\colon\Omega\rightarrow\mathbb{C}^3$ be a function $w=w_0+w_1e_1+w_2e_2$. We consider the solutions of $$ \left\{\begin{array}{rcll} z_2\left( \dfrac{\partial w_0}{\partial z_0}- \dfrac{\partial w_1}{\partial z_1}- \dfrac{\partial w_2}{\partial z_2} \right)+kw_2&=&0, & k\in\mathbb{R}\\[3mm] \dfrac{\partial w_1}{\partial\bar{z}_2}&=& \dfrac{\partial w_2}{\partial\bar{z}_1},\\[3mm] \dfrac{\partial w_0}{\partial\bar{z}_1}&=& -\dfrac{\partial w_1}{\partial\bar{z}_0},\\[3mm] \dfrac{\partial w_0}{\partial\bar{z}_2}&=& -\dfrac{\partial w_2}{\partial\bar{z}_0}. \end{array}\right. $$ Our system is a modification of a system introduced by Li Yucheng and Qiao Yuying and a complexification of a system introduced by H.\ Leutwiler. These $\mathrm{M}_k$-solutions are connected with $k$-hyperbolic harmonic functions $h$ by $$ w= \frac{\partial h}{\partial\bar{z}_0}- \frac{\partial h}{\partial\bar{z}_1}e_1- \frac{\partial h}{\partial\bar{z}_2}e_2, $$ where $h$ satisfies $$ z_2\Delta h-4k\frac{\partial h}{\partial\bar{z}_2}=0. $$ The $k$-hyperbolic harmonic functions are also connected to polyharmonic ones, $\Delta^kh$ is harmonic. We find basic properties for $\mathrm{M}_k$-solutions and $k$-hyperbolic harmonic functions and examine their bases.
Keywords: Modified quaternionic analysis, biquaternions, hyperbolic harmonic functions.
MSC 2000: Primary 30G35, 58G16; Secondary 30A05, 30F45.
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