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Split Octonionic Cauchy Integral Formula

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Abstract

Cauchy integral formulas for the quaternions have been known since 1926. This result has previously been extended to the octonions and split quaternions. These techniques are extended to the split octonions. Thus Cauchy integral formulas are now known for all composition algebras over the real numbers.

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References

  1. Baez, J.C.: The Octonions. Bull. Am. Math. Soc., 39:145–205 (2002). [Erratum: Bull. Am. Math. Soc.42,213(2005)]

    Article  MathSciNet  Google Scholar 

  2. Conway, J.H., Smith, D.A.: On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry. AK Peters, Natick (2003)

    Book  Google Scholar 

  3. Dentoni, P., Sce, M.: Funzioni regolari nell’algebra di Cayley. Rendiconti del Seminario Matematico della Università di Padova 50, 251–267 (1973)

    MathSciNet  MATH  Google Scholar 

  4. Dray, T., Manogue, C.A.: The Geometry of the Octonions. World Scientific, Singapore (2015)

    Book  Google Scholar 

  5. Emanuello, J.A.: Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries. Ph.D. Thesis, Florida State University (2015)

  6. Emanuello, J.A., Nolder, C.A.: Notions of regularity for functions of a split-quaternionic variable. In: Clifford Analysis and Related Topics, pp. 73–96. Springer International Publishing, Berlin (2018)

    Chapter  Google Scholar 

  7. Fueter, R.: Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen. Commentarii mathematici Helvetici, 8:371–378 (1935/36)

  8. Huerta, J.: Division Algebras, Supersymmetry and Higher Gauge Theory. Ph.D. thesis, UC, Riverside (2011)

  9. Li, X., Peng, L.: The Cauchy integral formulas on the octonions. Bull. Belg. Math. Soc. Simon Stevin 9(1), 47–64 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Libine, M.: Hyperbolic Cauchy integral formula for the split complex numbers (2008). arXiv:0712.0375

  11. Libine, M.: An invitation to split quaternionic analysis. In: Sabadini, I., Sommen, F. (eds.) Hypercomplex Analysis and Its Applications, pp. 161–180. Birkhäuser (2011). arXiv:1009.2540

    MATH  Google Scholar 

  12. McCarthy, J.M.: Introduction to Theoretical Kinematics. MIT Press, Cambridge (1990)

    Google Scholar 

  13. Shoemake, K.: Animating rotation with quaternion curves. SIGGRAPH Comput. Graph. 19(3), 245–254 (1985)

    Article  Google Scholar 

Download references

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  1. Florida State University, 208 Love Building, 1017 Academic Way, Tallahassee, FL, 32306-4510, USA

    Benjamin Prather

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  1. Benjamin Prather
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Correspondence to Benjamin Prather.

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Communicated by Swanhild Bernstein

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Prather, B. Split Octonionic Cauchy Integral Formula. Adv. Appl. Clifford Algebras 29, 89 (2019). https://doi.org/10.1007/s00006-019-1010-z

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