@article{Gryshchuk_2022, title={Monogenic functions with values in commutative complex algebras of the second rank with unity and the generalized biharmonic equation with double characteristics}, volume={74}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/6948}, DOI={10.37863/umzh.v74i1.6948}, abstractNote={<p>UDC 517.9<br><br>We prove that any two-dimensional algebra $\mathbb{B}_{\ast}$ of the second rank with unity over the field of complex numbers $\mathbb{C}$ contains basises $\{e_1,e_2\},$ for which the $\mathbb{B}_{\ast}$-valued ``analytic’’ functions $\Phi(xe_1+ye_2),$ where $x$ and $y$ are real variables, satisfy a homogeneous PDE of the fourth order with complex coefficients such that its characteristic equation has just one multiple root and the other roots are simple.<br>The set of all triples $(\mathbb{B}_{\ast}, \{e_1,e_2\}, \Phi)$ is described in the explicit form.<br><br></p>}, number={1}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Gryshchuk , S. V.}, year={2022}, month={Jan.}, pages={14 - 23} }