complex line

plural complex lines

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

   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.

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

   2 more 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.

   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.

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

   Synonyms: complex plane