ABSTRACT
From the point of view of complex analysis, harmonic functions in ℂ are particularly interesting as being at least locally the real parts of holomorphic functions. Only a subclass of the harmonic functions in ℂ n are locally the real parts of holomorphic functions when n > 1, and it is clearly of some importance to investigate more closely this subclass. It is a consequence of Hartogs’s theorem that holomorphic functions of several variables can be characterized as those functions that are holomorphic in each variable separately, but there is no corresponding characterization of the real parts of holomorphic functions. For example, the function u(z 1, z 2) = x 1 y 2 is the real part of a holomorphic function of each variable separately but is easily seen not to be the real part of a holomorphic function of two variables. However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in ℂ n through a point A ∈ ℂ n in the direction of a vector B ∈ ℂ n is the one-dimensional complex submanifold of ℂ n described parametrically as {A + tB: t ∈ C}.
