Skip to main content
Springer Nature Link
Log in

Biquaternion Extensions of Analytic Functions

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Here, “Cayley-Dickson” indicates an algebraic form, and is not necessarily related to Cayley-Dickson construction.

  2. 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, \(-\ldots -\ldots +\).

References

  1. Deavours, C.: The quaternion calculus. Am. Math. Mon. 80, 995–1008 (1973)

    Article  MathSciNet  Google Scholar 

  2. Imaeda, K.: A new formulation of classical electrodynamics. Il Nuovo Cimento B 32, 138–162 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  3. Fueter, R.: Die Funktionentheorie der Differentialgleichungen \(\Delta \)u= 0 und \(\Delta \)u= 0 mit vier reellen Variablen. Commentarii Mathematici Helvetici 7, 307–330 (1934)

    Article  MathSciNet  Google Scholar 

  4. Fueter, R.: Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen. Commentarii Mathematici Helvetici 8, 371–378 (1935)

    Article  MathSciNet  Google Scholar 

  5. Sangwine, S.J., Alfsmann, D.: Determination of the biquaternion divisors of zero, including the idempotents and nilpotents. Adv. Appl. Clifford Algebras 20, 401–410 (2010)

    Article  MathSciNet  Google Scholar 

  6. Ringleb, F.: Beiträge zur funktionentheorie in hyperkomplexen Systemen I. Rendiconti del Circolo Matematico di Palermo 57, 311–340 (1933)

    Article  Google Scholar 

  7. Futagawa, M.: On the theory of functions of a quaternary variable. Tohoku Math. J 29, 175–222 (1928)

    MATH  Google Scholar 

  8. Futagawa, M.: On the theory of functions of a quaternary variable (Part II). Tohoku Math. J. 35, 69–120 (1932)

    MATH  Google Scholar 

  9. Riley, J.D.: Contributions to the theory of functions of a bicomplex variable. Tohoku Math. J. 5, 132–165 (1953)

    Article  MathSciNet  Google Scholar 

  10. Luna-Elizarraras, M., Shapiro, M., Struppa, D., Vajiac, A.: Bicomplex numbers and their elementary functions. Cubo (Temuco) 14, 61–80 (2012)

    Article  MathSciNet  Google Scholar 

  11. Oba, R.: Eigen-decomposition of quaternions. Adv. Appl. Clifford Algebras 28, 94 (2018)

    Article  MathSciNet  Google Scholar 

  12. Sangwine, S.J.: Biquaternion (complexified quaternion) roots of -1. Adv. Appl. Clifford Algebras 16, 63–68 (2006)

    Article  MathSciNet  Google Scholar 

  13. Sangwine, S.J., Ell, T.A., le Bihan, N.: Fundamental representations and algebraic properties of biquaternions or complexified quaternions. Adv. Appl. Clifford Algebras 21, 607–636 (2011)

    Article  MathSciNet  Google Scholar 

  14. Unicode Consortium. 2018 CJK Symbols and Punctuation

  15. ISO. 2006 ISO/IEC 10967-3: Information technology — Language independent arithmetic — Part 3: Complex integer and floating point arithmetic and complex elementary numerical functions

  16. Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolic numbers. Anal. Univ. Oradea, fasc. Math. 11, 71–110 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Sangwine, S.J., le Bihan, N.: Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form. Adv. Appl. Clifford Algebras 20, 111–120 (2010)

    Article  MathSciNet  Google Scholar 

  18. Hitzer, E., Helmstetter, J., Abłamowicz, R.: Square roots of -1 in real Clifford algebras. Springer, In Quaternion and Clifford Fourier Transforms and Wavelets (2013)

    Book  Google Scholar 

  19. Sudbery, A.: Quaternionic analysis. Mathe. Proc. Cambridge Philosophical Soc. 85, 199–225 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  20. Ward, JP.: Quaternions and Cayley Numbers. Springer Science & Business Media(1997)

  21. Horn, RA., Johnson, CR.: Matrix Analysis. Cambridge University Press, second edition (2012)

  22. Price, G.B.: An Introduction to Multicomplex Spaces and Functions. Taylor & Francis (1991)

    MATH  Google Scholar 

  23. Rönn S.: Bicomplex algebra and function theory (2001) arXiv preprintarXiv:0101200 [math]

  24. Campos, H.M., Kravchenko, V.V.: Fundamentals of bicomplex pseudoanalytic function theory: Cauchy integral formulas, negative formal powers and Schrödinger equations with complex coefficients. Complex Analysis and Operator Theory 7, 485–518 (2013)

    Article  MathSciNet  Google Scholar 

  25. Woods, D.C., Bolton, J.S., Rhoads, J.F.: On the use of evanescent plane waves for low-frequency energy transmission across material interfaces. J. Acoustical Soc. Am. 138, 2062–2078 (2015)

    Article  ADS  Google Scholar 

  26. Markushevich, AI.: Theory of Functions of a Complex Variable. AMS Chelsea Publishing (1977)

  27. Landau, LD., Lifshitz, EM.: Fluid Mechanics, Course of Theoretical Physics vol. 6. Pergamon Press (1959)

  28. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press (1967)

    MATH  Google Scholar 

  29. Delgado, B.B., Porter, R.M.: General solution of the inhomogeneous div-curl system and consequences. Adv. Appl. Clifford Algebras 27, 3015–3037 (2017)

    Article  MathSciNet  Google Scholar 

  30. Bers, L.: Theory of Pseudo-Analytic Functions. New York University, Institute for Mathematics and Mechanics (1953)

    MATH  Google Scholar 

  31. Kravchenko, VV.: Applied Pseudoanalytic Function Theory. Birkhäuser Basel (2009)

Download references

Acknowledgements

This research is sponsored by the Office of Naval Research.

Author information

Authors and Affiliations

  1. Acoustics Division, Code 7167, Naval Research Laboratory, Washington, DC, 20375, USA

    Roger M Oba

Authors
  1. Roger M Oba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roger M Oba.

Ethics declarations

Conflict of interests

The author has no competing interests to declare that are relevant to the content of this article. This study created no new data; this article has no associated data.

Additional information

Communicated by Uwe Kaehler.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

DISTRIBUTION A: Approved for public release, distribution is unlimited.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oba, R.M. Biquaternion Extensions of Analytic Functions. Adv. Appl. Clifford Algebras 32, 47 (2022). https://doi.org/10.1007/s00006-022-01238-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-022-01238-8

Keywords

Mathematics Subject Classification

Search

Navigation