Complex coordinate space

{{short description|Space formed by the n-tuples of complex numbers}}

In mathematics, the n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers, also known as complex vectors. The space is denoted $\backslash Complex^n$, and is the n-fold Cartesian product of the complex line $\backslash Complex$ with itself. Symbolically,

$$\backslash Complex^n\; =\; \backslash left\backslash \{\; (z\_1,\backslash dots,z\_n)\; \backslash mid\; z\_i\; \backslash in\; \backslash Complex\backslash right\backslash \}$$

or

$$\backslash Complex^n\; =\; \backslash underbrace\{\backslash Complex\; \backslash times\; \backslash Complex\; \backslash times\; \backslash cdots\; \backslash times\; \backslash Complex\}\_\{n\}.$$

The variables $z\_i$ are the (complex) coordinates on the complex n-space.

The special case $\backslash Complex^2$, called the complex coordinate plane, is not to be confused with the complex plane, a graphical representation of the complex line.

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 $\backslash Complex^n$ with the 2n-dimensional real coordinate space, $\backslash mathbb\; R^\{2n\}$. With the standard Euclidean topology, $\backslash Complex^n$ 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.