A Correspondence of Hyperholomorphic and Monogenic Functions in ℝ⁴

Abstract

Let ” be the algebra of quaternions generated by e ₁, e ₂ satisfying e _i e _j + e _i e _j = -2δ _ij for i = 1,2. Consider the Clifford algebra $\mathbb{H} \oplus \mathbb{H}$ generated by e₁, e₂ and e₃. Any element x in $\mathbb{H} \oplus \mathbb{H}$ may be decomposed as x = Px + Qxe ₃ for quaternions Px and Qx. The Dirac operator in ℝ⁴ is defined by $D = \frac{\partial } {{\partial x_0 }} + \frac{\partial } {{\partial x_1 }}e1 + \frac{\partial } {{\partial x_2 }} + \frac{\partial } {{\partial x_3 }}e_3$ Leutwiler noticed that the power function $(x_0 + x_1 e_1 + x_2 e_2 + x_3 e_3 )^m$ is a solution of the modified Cauchy-Riemann system $x_3 Df + 2f_3 = 0$ which has connections to the hyperbolic metric. We study solutions of the equation $x_3 Df + 2Q'(f) = 0$ called hyperholomorhic functions, where ' is the main involution in $\mathbb{H} \oplus \mathbb{H}$. We prove that for any monogenic function g in $\mathbb{H} \oplus \mathbb{H}$ there exists locally a hyperholomorphic functions f with Δf = g. Moreover, if f is hyperholomorphic, then Δf is monogenic.