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Split Octonionic Cauchy Integral Formula

  • Benjamin PratherEmail author
Article
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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.

Keywords

Split Octonion Cauchy integral formula 

Mathematics Subject Classification

17A75 45E05 35L10 

Notes

References

  1. 1.
    Baez, J.C.: The Octonions. Bull. Am. Math. Soc., 39:145–205 (2002). [Erratum: Bull. Am. Math. Soc.42,213(2005)]MathSciNetCrossRefGoogle Scholar
  2. 2.
    Conway, J.H., Smith, D.A.: On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry. AK Peters, Natick (2003)CrossRefGoogle Scholar
  3. 3.
    Dentoni, P., Sce, M.: Funzioni regolari nell’algebra di Cayley. Rendiconti del Seminario Matematico della Università di Padova 50, 251–267 (1973)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dray, T., Manogue, C.A.: The Geometry of the Octonions. World Scientific, Singapore (2015)CrossRefGoogle Scholar
  5. 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)Google Scholar
  6. 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)CrossRefGoogle Scholar
  7. 7.
    Fueter, R.: Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen. Commentarii mathematici Helvetici, 8:371–378 (1935/36)Google Scholar
  8. 8.
    Huerta, J.: Division Algebras, Supersymmetry and Higher Gauge Theory. Ph.D. thesis, UC, Riverside (2011)Google Scholar
  9. 9.
    Li, X., Peng, L.: The Cauchy integral formulas on the octonions. Bull. Belg. Math. Soc. Simon Stevin 9(1), 47–64 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Libine, M.: Hyperbolic Cauchy integral formula for the split complex numbers (2008). arXiv:0712.0375
  11. 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 zbMATHGoogle Scholar
  12. 12.
    McCarthy, J.M.: Introduction to Theoretical Kinematics. MIT Press, Cambridge (1990)Google Scholar
  13. 13.
    Shoemake, K.: Animating rotation with quaternion curves. SIGGRAPH Comput. Graph. 19(3), 245–254 (1985)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Florida State UniversityTallahasseeUSA

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