Pluriharmonic function

Remarks

Equivalently, for all $a \in \Omega$ and $b \in \C^n$, the function $\lambda \mapsto u(a + \lambda b)$ is harmonic where it is defined. Example: $u = \operatorname{Re} f$ for a holomorphic $f$ is pluriharmonic, because $\partial f / \partial \bar z_k = 0$ and $\partial \bar f / \partial z_j = 0$ give $\partial^2 (f + \bar f) / \partial z_j \partial \bar z_k = 0$. Non-example: $u = \abs{z_1}^2$ has $\partial^2 u / \partial z_1 \partial \bar z_1 = 1 \ne 0$. Conversely, 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. A pluriharmonic function is real analytic. The condition can be written $\partial \bar\partial u = 0$.