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 a such function is referred as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined. However, in modern expositions of the theory of functions of several complex variables 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. 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 a such function is referred as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined. However, in modern expositions of the theory of functions of several complex variables 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. Definition 1. Let G ⊆ ℂn be a complex domain and f : G → ℂ be a C2 (twice continuously differentiable) function. The function f is called pluriharmonic if, for every complex line formed by using every couple of complex tuples a, b ∈ ℂn, the function is a harmonic function on the set 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. This article incorporates material from pluriharmonic function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.