# complex line

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## English[edit]

### Noun[edit]

**complex line** (*plural* **complex lines**)

- (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 through a point in the direction of a vector is the one-dimensional complex submanifold of described parametrically as .

- 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
**2014**, Paul M. Gauthier,*Lectures on Several Complex Variables*, Springer (Birkhäuser), page 39,- A
in is a set of the form , where and are fixed points in , with . Let us say that is the**complex line****complex line**through in the “direction” .

- A
**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 question of finding different familes

#### Usage notes[edit]

- 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.

#### Synonyms[edit]

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

#### Derived terms[edit]

#### Translations[edit]

affine subspace that is 1-dimensional over the complex numbers

### Further reading[edit]

- complex line on
*n*Lab