Про моногенні функції на розширеннях комутативної алгебри

Abstract

A relation between spatial potential fields and analytic functions given in commutative algebras was established by P.~W.~Ketchum who has shown that every analytic function $\Phi(\zeta)$ of the variable

$\zeta=xe_1+ye_2+ze_3$ satisfies the equation

\[ \Delta_3 u(x,y,z):=

\left(\frac{{\partial}^2}{{\partial x}^2}+

\frac{{\partial}^2}{{\partial y}^2}+

\frac{{\partial}^2}{{\partial z}^2}\right)u(x,y,z)

=0

\]

in the case where the elements $e_1, e_2, e_3$ of a commutative algebra satisfy the condition

\[ e_1^2+e_2^2+e_3^2=0\,,\]

because

\[% \begin{equation}\label{garm}

\frac{{\partial}^{2}\Phi}{{\partial x}^{2}}+

\frac{{\partial}^{2}\Phi}{{\partial y}^{2}}+

\frac{{\partial}^{2}\Phi}{{\partial z}^{2}}\equiv{\Phi}''(\zeta) \

(e_1^2+e_2^2+e_3^2)=0\,,\medskip

\] %\end{equation}

where $\Phi'':=(\Phi')'$ and $\Phi'(\zeta)$ is defined by the equality $d\Phi=\Phi'(\zeta)d\zeta$.

\apar

I.~P.~Mel'nichenko noticed that doubly differentiable in the sense of Gateaux functions form the largest algebra of functions $\Phi$ satisfying identically the above equalities, where $\Phi''$ is the Gateaux second derivative of function $\Phi$.

\apar

It is proved in [1] that for constructing solutions of the equation

\[ %\]\begin{equation}\label{dopolnenije----1+9}

\sum\limits_{\alpha+\beta+\gamma=N}C_{\alpha,\beta,\gamma}\,

\frac{\partial^N U} {\partial x^\alpha\,\partial y^\beta\,\partial

z^\gamma}=0, \quad C_{\alpha,\beta,\gamma}\in\mathbb{R}.

\] %\end{equation}

in the form of components of monogenic functions with values in finite-dimensional commutative associative algebras it suffices to confine itself to studying monogenic functions in algebras of a certain type, that is, in algebras $\mathbb{A}_n$.

\apar

For $n$-dimensional $(2\leq n<\infty)$ commutative associative algebra $\mathbb{A}_n$ we introduce a concept of \textit{expansion} as a set of $(n+1)$-dimensional commutative associative algebras with certain multiplication rules.

A relation between monogenic (continuous and differentiable in the sense of G\^{a}teaux) functions in the algebra $\mathbb{A}_n$ and monogenic functions that defined on expansions of $\mathbb{A}_n$ is established.

For the equation above it will mean the following: if its complex-valued solution $U(x, y, z)$ is a component of monogenic function in the algebra $ \mathbb{A}_n$ for $ n <N $, then there exists the algebra of the form $\mathbb{A}_N$ and there exists a monogenic function $\Phi$ in $\mathbb{A}_N$ such that $ U (x, y, z) $ is a component of the function $\Phi$.

The results obtained in this paper hold for the equations of the above type of $d$ variables for all integer $2\leq d\leq2N$.

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