complex line

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.