complex line

Definitions

A 1-dimensional affine subspace of a vector space over the complex numbers.

analytic

Examples

“1990, R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Volume 1 Function Theory, Wadsworth & Brooks/Cole, page 102, 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.”

“A complex line in #92;Cⁿ is a set⟳ of the form⟳ l#61;#92;#123;z#58;z#61;a#43;#92;lambdab,#92;lambda#92;in#92;C#92;#125;, where a and b are fixed points in #92;Cⁿ, with b#92;ne 0. Let⟳ us say⟳ that l is the complex line through a in the “direction” b.”

“The question⟳ of finding different familes^([sic]) of complex lines sufficient for holomorphic extension was put⟳ in [12]. Clearly, the family of complex lines passing through one point⟳ is not enough. As shown in [16], the family of complex lines passing through a finite number⟳ of points also, generally speaking, is not sufficient.”

CEFR level