Hypermonogenic function in Clifford Analysis

摘要

Let A_n(R) be a real Clifford algebra over a n+1-dimensional real vector space R~(n+1) with orthogonal basis e := {e_0, e_1, …, e_n}, where e_0 = 1 is a unit element in R~(n+1). Then A~n(R) has its basis e_0, e_1, …, e_n; e_1e_2, …, e_(n-1)e_n; …; e_1…, e_n. Hence an arbitrary element of the basis may be written as e_A = e_(α1),… ,e_(αh), here A = {α_1, …,(α_h} is contained in {1, … ,n} and 1 ≤ α_1 ＜ α_2 ＜ …＜ α_h ≤ n and when A = O (empty set) e_A = e_0. So real Clifford algebra is composed by the elements having the type α = Σ_Ax_ae_A, in which x_A(∈ R) are real numbers (see [1]). In general, one has e_i~2 =-1, i = 1, …, n, and e_ie_j + e_je_i = 0, i,j = 1, … , n, i ≠ j. Noting that the real vector space R~(n+1) consists of the element z := x_0e_0 + … + x_ne_n,we can consider that the elements z := x_0e_0 + …+ x_2e_n and z = (x_0, … , x_n) are identical, and denote by Rez = X_0 the real part x_0 of z.