Abstract
This paper presents a three-step program for extension of functions of complex analysis to the biquaternions by means of Cauchy’s integral formula: I. Investigate biquaternion bases consisting of roots of \(-1\). A complex valued standard function (standardization factor) determines roots of \(-1\). A root of \(-1\) with a non-zero imaginary part, can uniquely determine a biquaternion ortho-standard basis. II. A single reference basis element determines two subspaces, one the span of scalars and the reference element, the other pure vector biquaternions orthogonal to the reference. The subspaces represent the distinct parts of the generalized Cayley-Dickson form. The Peirce decomposition projects into two subspaces: one is the span of the related idempotents and the other of the nilpotents. III. Using invertible elements in each of these subspaces, biquaternion functional extensions of holomorphic functions follow by Cauchy’s integral formula. Extensions retain analyticity in each biquaternion component. Cauchy integral formula uses separate idempotent and nilpotent representations of biquaternion reciprocals to define holomorphic function extensions. The Peirce projections allow extension to all viable biquaternions.
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Notes
Here, “Cayley-Dickson” indicates an algebraic form, and is not necessarily related to Cayley-Dickson construction.
N.B. The sign pairs in a sentence, such as \(\pm \dots \pm \dots \mp \) for example, give a pair of statements: the first with the top signs, \(+\ldots +\ldots -\), and a second with the lower signs, \(-\ldots -\ldots +\).
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This research is sponsored by the Office of Naval Research.
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Oba, R.M. Biquaternion Extensions of Analytic Functions. Adv. Appl. Clifford Algebras 32, 47 (2022). https://doi.org/10.1007/s00006-022-01238-8
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DOI: https://doi.org/10.1007/s00006-022-01238-8
Keywords
- Biquaternion
- Biquaternion function theory
- Biquaternion integration
- Idempotent
- Nilpotent
- Peirce decomposition
- Cauchy integration