Complex coordinate space
In mathematics, the n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers. It is denoted [math]\displaystyle{ \Complex^n }[/math], and is the n-fold Cartesian product of the complex plane [math]\displaystyle{ \Complex }[/math] with itself. Symbolically, [math]\displaystyle{ \Complex^n = \left\{ (z_1,\dots,z_n) \mid z_i \in \Complex\right\} }[/math] or [math]\displaystyle{ \Complex^n = \underbrace{\Complex \times \Complex \times \cdots \times \Complex}_{n}. }[/math] The variables [math]\displaystyle{ z_i }[/math] are the (complex) coordinates on the complex n-space.
Complex coordinate space is a vector space over the complex numbers, with componentwise addition and scalar multiplication. The real and imaginary parts of the coordinates set up a bijection of [math]\displaystyle{ \Complex^n }[/math] with the 2n-dimensional real coordinate space, [math]\displaystyle{ \mathbb R^{2n} }[/math]. With the standard Euclidean topology, [math]\displaystyle{ \Complex^n }[/math] is a topological vector space over the complex numbers.
A function on an open subset of complex n-space is holomorphic if it is holomorphic in each complex coordinate separately. Several complex variables is the study of such holomorphic functions in n variables. More generally, the complex n-space is the target space for holomorphic coordinate systems on complex manifolds.
See also
References
- Gunning, Robert; Hugo Rossi, Analytic functions of several complex variables
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