Pluriharmonic function

In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined.^[1] However, in modern expositions of the theory of functions of several complex variables^[2] it is preferred to give an equivalent formulation of the concept, by defining pluriharmonic function a complex valued function whose restriction to every complex line is a harmonic function with respect to the real and imaginary part of the complex line parameter.

Formal definition

Definition 1. Let G ⊆ Cⁿ be a complex domain and f : G → R be a C² (twice continuously differentiable) function. The function f is called pluriharmonic if, for every complex line

${\displaystyle \{a+bz\mid z\in \mathbb {C} \}\subset \mathbb {C} ^{n}}$

formed by using every couple of complex tuples a, b ∈ Cⁿ, the function

${\displaystyle z\mapsto f(a+bz)}$

is a harmonic function on the set

${\displaystyle \{z\in \mathbb {C} \mid a+bz\in G\}\subset \mathbb {C} .}$

Definition 2. Let M be a complex manifold and f : M → R be a C² function. The function f is called pluriharmonic if

${\displaystyle dd^{c}f=0.}$

Basic properties

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.

See also

• Plurisubharmonic function
• Wirtinger derivatives