The Complex Clifford Analysis In Super-regular Functions

In this paper, resorting to the hypermonogenic function in real Clifford analysis , we define the hypermonogenic function in complex Clifford analysis and give the sufficient and necessary conditions of complex monogenic and complex hypermonogenic functions. The result is similar to the Cauchy-Riemann condition of complex analysis. So we get some relations between the real and complex Clifford functions. In addition, we discuss some properties of complex hypermonogenic functions.Theorem 1 f(z) is a complex Clifford function, its real part and imaginary part are real Clifford function u(z) v(z), then the sufficient and necessary conditions for f(z) = u(z)+iv(z) to be a complex monogenic function isDx is a real Dirac operator about x, Dy is a real Dirac operator about yIt is similar to the C-R conditions of single complex function, but they are different.Theorem 2 fi is an open subset of Cn+1, is com-plex hypermonogenic functions, then are all complexhypermonogenic functions.Theorem 3 The sufficient and necessary conditions for An(C) to be complex hypermonogenic functions are:Here, Mx, My are given by (3) in main paper.