Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic

Abstract

The statement that any two-dimensional algebra $\mathbb{B}_{\ast}$ of the second rank with unity over the field of complex numbers contains such a basis $\{e_{1},e_{2}\}$ that $\mathbb{B}_{\ast}$-valued ''analytic'' functions $\Phi(xe_{1}+ye_{2})$ ($x$, $y$ are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation has a triple root is proved. A set of all triples $\left( \mathbb{B}_{\ast},\{e_{1},e_{2}\},\Phi\right)$ is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these ''analytic'' functions.