Pluriharmonic function

Let

:$f\; colon\; G\; subset\; \{mathbb\{C^n\; o\; \{mathbb\{C$

be a $C^2$ (twice continuously differentiable) function. $f$ is called pluriharmonic if for every complex line

:$\{\; a\; +\; b\; z\; mid\; z\; in\; \{mathbb\{C\; \}$

the function

:$z\; mapsto\; f(a\; +\; bz)$

is a harmonic function on the set

:$\{\; z\; in\; \{mathbb\{C\; mid\; a\; +\; b\; z\; in\; G\; \}$.

Notes

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. However a function being harmonic in each variable separately does not imply that it is pluriharmonic.

Bibliography

* Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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