Clifford analysis

Remarks

The anticommutation relations give $D^2 = -\Delta$, where $\Delta$ is the Laplacian of $\R^n$: the mixed terms $e_j e_k \partial_j \partial_k + e_k e_j \partial_k \partial_j$ cancel. So every monogenic function of class $C^2$ has harmonic components.

Example: for $n = 1$, $\mathbb{R}_{0,1} = \R + \R e_1$ with $e_1^2 = -1$ is a copy of $\C$, and $D = e_1 \partial_x$ only sees one real variable. To recover complex analysis one uses instead the operator $\partial_{x_0} + e_1 \partial_{x_1}$ on $\R^2$, which is twice the Wirtinger derivative $\partial_{\bar z}$; many authors work with this operator $\partial_{x_0} + \sum_j e_j \partial_{x_j}$ on $\R^{n+1}$.

Non-example: harmonic analysis of real-valued functions alone is not Clifford analysis, since it uses only $\Delta$ and not its first-order factor $D$.