complex line

complex line (plural complex lines)

1. (complex analysis, analytic geometry) A 1-dimensional affine subspace of a vector space over the complex numbers.
   • 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 ${\displaystyle \mathbb {C} ^{n}}$ through a point ${\displaystyle A\in \mathbb {C} ^{n}}$ in the direction of a vector ${\displaystyle B\in \mathbb {C} ^{n}}$ is the one-dimensional complex submanifold of ${\displaystyle \mathbb {C} ^{n}}$ described parametrically as ${\displaystyle \{A+tB:t\in \mathbb {C} \}}$ .

   • 2014, Paul M. Gauthier, Lectures on Several Complex Variables, Springer (Birkhäuser), page 39,

     A complex line in ${\displaystyle \mathbb {C} ^{n}}$ is a set of the form ${\displaystyle l=\{z:z=a+\lambda b,\lambda \in \mathbb {C} \}}$ , where ${\displaystyle a}$ and ${\displaystyle b}$ are fixed points in ${\displaystyle \mathbb {C} ^{n}}$ , with ${\displaystyle b\neq 0}$ . Let us say that ${\displaystyle l}$ is the complex line through ${\displaystyle a}$ in the “direction” ${\displaystyle b}$ .

   • 2018, Bairambay Otemuratov A Mulitidimensional Analogue of Hartogs's Theorem on n-Circular Domains for Integrable Functions, Zair Ibragimov, Norman Levenberg, Utkir Rozikov, Azimbay Sadullaev (editors), Algebra, Complex Analysis, and Pluripotential Theory: 2 USUZCAMP, 2017, Springer, page 110,

     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.

• The term emphasises the structure's 1-dimensional aspect. The structure is 1-dimensional strictly in the sense that it is defined (as a vector space) over a single dimension of the complex numbers: topologically, it is equivalent to the real plane.

• (affine subspace that is 1-dimensional over the complex numbers): complex plane

• complex line bundle

• complex line on nLab