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Quaternionic Analysis and Some Conventional Theories

In: Operator Theory

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  • Michael Shapiro

    (Escuela Superior de Física y Matemáticas, del Instituto Politécnico Nacional, Departamento de Matemáticas)

Abstract

Firstly, it is recalled briefly what are the analogues of the usual Cauchy–Riemann operators of complex analysis for quaternion-valued functions. Two of them, the Fueter and the Moisil–Théodoresco operators, are related to the Laplace equation, and one more is related to the Helmholtz operator. Secondly, it is shown that each of the following four theories can be embedded into one of the above quaternionic ones: holomorphic functions in ℂ 2 $$\mathbb{C}^{2}$$ , vector analysis, time-harmonic electromagnetic, and time-harmonic spinor fields. It is illustrated with specific examples that such embeddings prove to be rather fruitful for the conventional theories.

Suggested Citation

  • Michael Shapiro, 2015. "Quaternionic Analysis and Some Conventional Theories," Springer Books, in: Daniel Alpay (ed.), Operator Theory, edition 127, chapter 49, pages 1423-1446, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-0667-1_25
    DOI: 10.1007/978-3-0348-0667-1_25
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