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

The statement that any two-dimensional algebra 𝔹 * of the second rank with unity over the field of complex numbers contains such a basis { e 1 ; e 2 } that 𝔹 * -valued “analytic” functions Φ( xe 1 + ye 2 ) ( x , y are real variables) satisfy such a fourth-order homogeneous partial differential equat...

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Bibliographic Details
Published inJournal of mathematical sciences (New York, N.Y.) Vol. 262; no. 2; pp. 154 - 164
Main Author Gryshchuk, Serhii V.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2022
Springer
Springer Nature B.V
Subjects
Algebra
Biharmonic equations
Complex numbers
Differential equations
Eigenvalues
Eigenvectors
Mathematical analysis
Mathematics
Mathematics and Statistics
Partial differential equations
Real variables
monogenic function
Associative commutative Banach algebra
fourth-order partial differential equation with complex coefficients
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Summary:The statement that any two-dimensional algebra 𝔹 * of the second rank with unity over the field of complex numbers contains such a basis { e 1 ; e 2 } that 𝔹 * -valued “analytic” functions Φ( 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 (𝔹 * ; { e 1 ; e 2 } ; 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.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-022-05807-x