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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be an open set.\nA function $u \\colon \\Omega \\to \\R$ of class $C^2$ is *pluriharmonic* if\n$$\\frac{\\partial^2 u}{\\partial z_j \\partial \\bar z_k} = 0 \\quad \\text{on } \\Omega \\text{ for all } j, k \\in \\set{1, \\dots, n},$$\nwhere $\\partial / \\partial z_j = \\frac{1}{2}(\\partial / \\partial x_j - i \\, \\partial / \\partial y_j)$ and $\\partial / \\partial \\bar z_k = \\frac{1}{2}(\\partial / \\partial x_k + i \\, \\partial / \\partial y_k)$ for $z_j = x_j + i y_j$.\n\n## Remarks\n\nEquivalently, for all $a \\in \\Omega$ and $b \\in \\C^n$, the function $\\lambda \\mapsto u(a + \\lambda b)$ is [[harmonic|0DEB0JMN0W]] where it is defined.\nExample: $u = \\operatorname{Re} f$ for a holomorphic $f$ is pluriharmonic, because $\\partial f / \\partial \\bar z_k = 0$ and $\\partial \\bar f / \\partial z_j = 0$ give $\\partial^2 (f + \\bar f) / \\partial z_j \\partial \\bar z_k = 0$.\nNon-example: $u = \\abs{z_1}^2$ has $\\partial^2 u / \\partial z_1 \\partial \\bar z_1 = 1 \\ne 0$.\nConversely, on a ball, or more generally on a simply connected domain, every pluriharmonic function is the real part of a holomorphic function; on other domains this holds locally.\nA pluriharmonic function is real analytic.\nThe condition can be written $\\partial \\bar\\partial u = 0$.\n",
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