# pluriharmonic function

###### Definition.

Let $f\colon G\subset{\mathbb{C}}^{n}\to{\mathbb{C}}$ be a $C^{2}$ (twice continuously differentiable) function. $f$ is called pluriharmonic if for every complex line $\{a+bz\mid z\in{\mathbb{C}}\}$ the function $z\mapsto f(a+bz)$ is harmonic on the set $\{z\in{\mathbb{C}}\mid a+bz\in G\}$.

Note that every pluriharmonic function is a harmonic function, but not the other way around. Further it can be shown that for holomorphic functions of several complex variables the real (and the imaginary) parts are locally pluriharmonic functions. Do note however that just because a function is harmonic in each variable separately does not imply that it is pluriharmonic.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title pluriharmonic function PluriharmonicFunction 2013-03-22 14:29:01 2013-03-22 14:29:01 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 31C05 msc 32A50 msc 31C10 pluriharmonic