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On some structural sets and a quaternionic ( φ , ψ ) ‐hyperholomorphic function theory
Author(s) -
Abreu Blaya Ricardo,
Bory Reyes Juan,
Guzmán Adán Alí,
Kaehler Uwe
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300072
Subject(s) - mathematics , quaternion , pure mathematics , cauchy–riemann equations , cauchy distribution , function (biology) , field (mathematics) , riemann hypothesis , set (abstract data type) , algebra over a field , mathematical analysis , geometry , evolutionary biology , biology , computer science , programming language
Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy‐Riemann equations to the quaternion skew field H . It relies heavily on results on functions defined on domains inR 4 ( orR 3 ) with values in H . This theory is centred around the concept of ψ‐hyperholomorphic functions related to a so‐called structural set ψ ofH 4 ( orH 3 ) respectively. The main goal of this paper is to develop the nucleus of the ( φ , ψ ) ‐hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy‐Riemann operators associated to a pair φ , ψ of structural sets ofH 4 . Following a matrix approach, a generalized Borel‐Pompeiu formula and the corresponding Plemelj‐Sokhotzki formulae are established.