Regular functions with values in ternary number system on the complex Clifford analysis. (English) Zbl 1295.30113

Summary: We define a new modified basis \(\hat{i}\) which is an association of two bases, \(e_1\) and \(e_2\). We give an expression of the form \(z=x_0+\hat{i}\overline{z_0}\), where \(x_0\) is a real number and \(\overline{z_0}\) is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.


30G35 Functions of hypercomplex variables and generalized variables
Full Text: DOI


[1] Fueter, R., Die funktionentheorie der differentialgleichungen \(\Delta u = 0\) und \(\Delta \Delta u = 0\) mit vier reellen variablen, Commentarii Mathematici Helvetici, 7, 1, 307-330 (1935) · Zbl 0012.01704 · doi:10.1007/BF01292723
[2] Deavours, C. A., The quaternion calculus, The American Mathematical Monthly, 80, 995-1008 (1973) · Zbl 0282.30040 · doi:10.2307/2318774
[3] Sudbery, A., Quaternionic analysis, Mathematical Proceedings of the Cambridge Philosophical Society, 85, 2, 199-224 (1979) · Zbl 0399.30038 · doi:10.1017/S0305004100055638
[4] Naser, M., Hyperholomorphic functions, Silberian Mathematical Journal, 12, 959-968 (1971) · Zbl 0247.30042
[5] Koriyama, H.; Mae, H.; Nôno, K., Hyperholomorphic functions and holomorphic functions in quaternionic analysis, Bulletin of Fukuoka University of Education, 60, 1-9 (2011)
[6] Nôno, K., Hyperholomorphic functions of a quaternion variable, Bulletin of Fukuoka University of Education, 32, 21-37 (1983) · Zbl 0511.30038
[7] Cho, E., De moivre’s formula for quaternions, Applied Mathematics Letters, 11, 6, 33-35 (1998) · Zbl 0938.11055 · doi:10.1016/S0893-9659(98)00098-6
[8] Sangwine, S. J.; Bihan, N. L., Quaternion polar representation with a complex modulus and complex argument inspired by the cayley-dickson form, Advances in Applied Clifford Algebras, 20, 1, 111-120 (2010) · Zbl 1223.16005 · doi:10.1007/s00006-008-0128-1
[9] Fueter, R., Die theorie der regulären funktionen einer quaternionenvariablen, Comptès Rendus du Congrès International des Mathèmaticiens, Oslo 1936, 1, 75-91 (1935)
[10] Brackx, F.; Delanghe, R.; Sommen, F., Clifford Analysis. Clifford Analysis, Research Notes in Mathematics, 76 (1982) · Zbl 0529.30001
[11] Lim, S. J.; Shon, K. H., Hyperholomorphic fucntions and hyperconjugate harmonic functions of octonion variables, Journal of Inequalities and Applications, 77, 1-8 (2013) · Zbl 1281.30035
[12] Lim, S. J.; Shon, K. H., Dual quaternion functions and its applications, Journal of Applied Mathematics, 2013 (2013) · Zbl 1397.30039 · doi:10.1155/2013/583813
[13] Lim, S. J.; Shon, K. H., Regularity of functions with values in a non-commutative algebra of complex matrix algebras
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.