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 𝔹* of the second rank with unity over the field of complex numbers contains such a basis {e1; e2} that 𝔹*-valued “analytic” functions Φ(xe1 + ye2) (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 (𝔹*; {e1; e2}; Φ) is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these “analytic” functions.