Monogenic function

Remarks

Example: identify $x \in \R^n$ with $\sum_j x_j e_j$, so that $x^2 = -\abs{x}^2$. The Cauchy kernel $E(x) = \dfrac{\bar x}{\abs{x}^n}$ with $\bar x = -x$ is left and right monogenic on $\R^n \setminus \set{0}$. A simpler example is $f(x) = x_1 e_1 - x_2 e_2$, since $Df = e_1 e_1 - e_2 e_2 = -1 + 1 = 0$.

Non-example: $f(x) = x = \sum_j x_j e_j$ is not monogenic, since $Df = \sum_j e_j^2 = -n \ne 0$.

Since $D^2 = -\Delta$, with $\Delta$ the Laplacian, each real component of a left monogenic function of class $C^2$ is harmonic. Many authors use instead the generalized Cauchy–Riemann operator $\partial_{x_0} + \sum_{j=1}^{n} e_j \partial_{x_j}$ on $\R^{n+1}$; for $n = 1$ it gives back holomorphic functions.