Abstract
This paper defines a hyperholomorphic function and a split harmonic function with values in split quaternions, expresses polar coordinate forms for split quaternions, and obtains some split Cauchy–Riemann systems that are equivalent. The paper provides split hyperholomorphic mappings on \({\Omega\subset \mathbb{C}^2}\) and researches the properties of split hyperholomorphic mappings.
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References
Cho E.: De moivre’s formula for quaternions. Appl. Math. Lett. 11(6), 33–35 (1998)
Carmody K.: Circular and hyperbolic quaternions, octonions and sedenions. Appl. Math. Comput. 28(1), 47–72 (1988)
Carmody K.: Circular and hyperbolic quaternions, octonions and sedenions—further results. Appl. Math. Comput. 84(1), 27–47 (1997)
Deavous C.A.: The quaternion calculus. Am. Math. Mon. 80(9), 995–1008 (1973)
Gotô S., Nôno K.: Regular functions with values in a commutative subalgebra \({\mathbb{C}(\mathbb{C})}\) of matrix algebra \({M(4;\mathbb{R})}\). Bull. Fukuoka Univ. Ed. Part III 61, 9–15 (2012)
Kajiwara, J., Li, X.D., Shon, K.H.: Regeneration in complex, quaternion and Clifford analysis. In: International colloquium on finite or infinite dimensional complex analysis and its applications. 2, Kluwer Academic Publishers, Vietnam, 2004
Kajiwara, J., Li, X.D., Shon, K.H.: Function spaces in complex and Clifford analysis. In: International colloquium on finite or infinite dimensional complex analysis and its applications. 14, Hue University, Vietnam, 2006
Kim, J.E., Lim, S.J., Shon, K.H.: Regular functions with values in ternary number system on the complex Clifford analysis. Abstr. Appl. Anal. 2013, Article ID. 136120, 7 (2013)
Kim, J.E., Lim, S.J., Shon, K.H.: Regularity of functions on the reduced quaternion field in Clifford analysis. Abstr. Appl. Anal. 2014, Article ID. 654798, 8 (2014)
Kim, J.E., Shon, K.H.: The regularity of functions on dual split quaternions in Clifford analysis. Abstr. Appl. Anal. 2014, Article ID. 369430, 8 (2014)
Kula L., Yayli Y.: Split quaternions and rotations in semi Euclidean space E 4. J. Korean Math. Soc. 44(6), 1313–1327 (2007)
Lang, S.: Calculus of Several Variables. Springer-Verlag, New York (1987)
Lim S.J., Shon K.H.: Hyperholomorphic fucntions and hyperconjugate harmonic functions of octonion variables. J. Inequal. Appl. 77, 1–8 (2013)
Lim, S.J., Shon, K.H.: Dual quaternion functions and its applications. J. Appl. Math. Article ID. 583813, 6 (2013)
Obolashvili E.: Some partial differential equations in Clifford analysis. Banach Cent. Publ. 37(1), 173–179 (1996)
Özdemir M., Ergin A.A.: Rotations with unit timelike quaternions in Minkowski 3-space. J. Geom. Phys. 56(2), 322–336 (2006)
Sangwine S.J., Bihan N.L.: Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley–Dickson form. Adv. Appl. Cliff. Algs 20(1), 111–120 (2010)
Wuming L.: Hyperbolic Euler formula in the n-dimensional Minkowski space. Adv. Appl. Cliff. Algs. 12(1), 7–11 (2002)
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Kim, J.E., Shon, K.H. Polar Coordinate Expression of Hyperholomorphic Functions on Split Quaternions in Clifford Analysis. Adv. Appl. Clifford Algebras 25, 915–924 (2015). https://doi.org/10.1007/s00006-015-0541-1
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DOI: https://doi.org/10.1007/s00006-015-0541-1