partial derivatives (with respect to $x$ and $y$, every element in $L_I$ being uniquely expressible as $x + yI$) and satisfies $$\overline{\partial}_I f (x + yI) := \frac 12\left(\frac {\partial f_I (x + yI )}{\partial x}
       + I \frac {\partial f_I (x + yI )}{\partial y}\right) = 0\; .
\label{leftslicereg}$$ Similarly, it is said to be slice right regular if $$\overline{\partial}_I f (x + yI) := \frac 12\left(\frac {\partial f_I (x + yI )}{\partial x}
       +  \frac {\partial f_I (x + yI )}{\partial y}I\right) = 0\; .
\label{rightslicereg}$$

With this definition all monomials of the form $a{\mathbf{q}}^n,~a\in H,~n\in\mathbb{N}$, are slice right regular while those of the form ${\mathbf{q}}^n a,\; a\in H,~n\in\mathbb{N}$, are slice left regular . Since regularity respects addition, all polynomials of the form $f({\mathbf{q}})=\sum_{t=0}^{n}a_t{\mathbf{q}}^t$, with $a_t\in H$, are slice right regular and similarly polynomials of the form $f({\mathbf{q}})=\sum_{t=0}^{n}{\mathbf{q}}^t a_t$, are slice left regular. Further, an analog of Abel’s theorem guarantees convergence of appropriate infinite power series.

[@Gra1]\[P1\] For any non-real quaternion ${\mathbf{q}}\in H-\mathbb{R}$, there exist, and are unique, $x,y\in\mathbb{R}$ with $y>0$, and $I\in\mathbb{S}$ such that ${\mathbf{q}}=x+yI$.

Henceforth we shall refer to slice right (left) regular functions simply as right (left) regular functions. We now define the Cullen derivative 