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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\mathbb{R}_{0,n}$ be the real [[Clifford algebra|A38B1QFK20]] generated by $e_1, \\ldots, e_n$ with $e_j^2 = -1$ and $e_j e_k = -e_k e_j$ for $j \\ne k$.\nLet $D = \\sum_{j=1}^{n} e_j \\dfrac{\\partial}{\\partial x_j}$ be the [[Dirac operator|3DXGJTEQZH]] on $\\R^n$.\nLet $U \\subset \\R^n$ be open and let $f \\colon U \\to \\mathbb{R}_{0,n}$ have continuous first [[partial derivatives|09EGC6F8PT]].\nThe function $f$ is *left monogenic* on $U$ if\n$$D f = \\sum_{j=1}^{n} e_j \\frac{\\partial f}{\\partial x_j} = 0 \\quad \\text{on } U.$$\nIt is *right monogenic* if $\\sum_{j} \\dfrac{\\partial f}{\\partial x_j} e_j = 0$ on $U$.\n\n## Remarks\n\nExample: identify $x \\in \\R^n$ with $\\sum_j x_j e_j$, so that $x^2 = -\\abs{x}^2$.\nThe Cauchy kernel $E(x) = \\dfrac{\\bar x}{\\abs{x}^n}$ with $\\bar x = -x$ is left and right monogenic on $\\R^n \\setminus \\set{0}$.\nA simpler example is $f(x) = x_1 e_1 - x_2 e_2$, since $Df = e_1 e_1 - e_2 e_2 = -1 + 1 = 0$.\n\nNon-example: $f(x) = x = \\sum_j x_j e_j$ is not monogenic, since $Df = \\sum_j e_j^2 = -n \\ne 0$.\n\nSince $D^2 = -\\Delta$, with $\\Delta$ the [[Laplacian|0E0B6RA8K0]], each real component of a left monogenic function of class $C^2$ is harmonic.\nMany 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.\n",
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