RELATIONS BETWEEN QUATERNIONIC DIFFERENTIAL AND THE CORRESPONDING CAUCHY RIEMANN SYSTEM

Abstract

Abstract. In this paper, we investigate several properties of quaternionicfunctions. We research some di erentials of quaternionic functions, andrelations between the di erentials and the corresponding Cauchy Riemannsystem in Cli ord analysis. 1. IntroductionThe quaternion eld Tis a non-commutative four dimensional skew eld ofreal numbers. The eld Tis identi ed with C 2 and R 4 , where R denotes the eld of real numbers and C denotes the eld of complex numbers.Naser [9] have studied some properties of hyperholomorphic functions overthe eld Tand quaternionic conjugate harmonic functions in 1971. In 1995,N^ono [10] has shown several properties of regular hypercomplex functions ontwo approaches. And N^ono [10] has searched properties of hyperholomorphicfunctions by using partial di erential equation.In 2011, Luna-Elizarraras and Shapiro [1] have shown that some propertiesfor regular functions in one complex analysis are feasible in Cli ord analysis.Also Luna-Elizarraras and Shapiro [1] have explained the derivatives of func-tions in one complex variable theory and the quaternionic analysis. In 2011,Koriyama et al. [8] have given regularities on quaternionic functions for severaldi erential operators in Cli ord analysis. Also Koriyama et al. [8] have shownthe corresponding Cauchy Riemann system for each operator D