a straightforward way.

(Cauchy-Fueter equations) [@Am; @Gra1]\[def-cauchy-fueter\] Let $f:H\longrightarrow H$ be a quaternion valued function of a quaternionic variable. We say that $f$ is left-regular if it satisfies the Cauchy-Fueter equation $$\frac{\partial_l f}{\partial{\overline{{\mathbf{q}}}}}=\frac{\partial f}{\partial x_0}+ i\frac{\partial f}{\partial x_1}+ j\frac{\partial f}{\partial x_2}+ k\frac{\partial f}{\partial x_3}=0 ,$$ and that $f$ is right-regular if it satisfies the other Cauchy-Fueter equation $$\frac{\partial_r f}{\partial{\overline{{\mathbf{q}}}}}=\frac{\partial f}{\partial x_0}+\frac{\partial f}{\partial x_1}i+\frac{\partial f}{\partial x_2}j+\frac{\partial f}{\partial x_3}k=0 .$$

Using this definition a theory of regular functions has been developed as a well structured theory. However, the unpleasant feature of this definition is that under the Cauchy-Fueter equations the function $f({\mathbf{q}})={\mathbf{q}}$ is not regular, and thereby none of the monomials or polynomials are regular. There have been several attempts to overcome this feature. The most promising, and recent attempt has appeared in [@Gra1] (see also [@Gra2]) where the following definition is offered. Let $\mathbb{S}=\{{\mathbf{q}}=x_1i+x_2j+x_3j~\vert~x_1,x_2,x_3\in\mathbb{R},~x_1^2+x_2^2+x_3^2=1\}$.

(Slice-regular functions [@Gra1])\[D2\] Let $\Omega$ be a domain in $H$. A real differentiable (i.e., with respect to $x_0$ and the $x_i,\; i=1,2,3$) function $f:\Omega\longrightarrow H$ is said to be slice left regular if, for every quaternion $I\in\mathbb{S}$, the restriction of $f$ to the complex line $L_I=\mathbb{R}+I\mathbb{R}$ passing through the origin, and containing $1$ and $I$, has continuous 