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Quaternionic Analysis and Some Conventional Theories

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Operator Theory

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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 \(\mathbb{C}^{2}\), 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.

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Authors and Affiliations

  1. Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Mexico City, Mexico

    Michael Shapiro

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  1. Michael Shapiro
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Correspondence to Michael Shapiro .

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  1. Ben-Gurion University of the Negev, Department of Mathematics, Beer-Sheva, Israel

    Daniel Alpay

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Shapiro, M. (2014). Quaternionic Analysis and Some Conventional Theories. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_25-1

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  • DOI: https://doi.org/10.1007/978-3-0348-0692-3_25-1

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  • Online ISBN: 978-3-0348-0692-3

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