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 ${\displaystyle \mathbb {C} ^{n}}$, and is the n-fold Cartesian product of the complex plane ${\displaystyle \mathbb {C} }$ with itself. Symbolically,

$${\displaystyle \mathbb {C} ^{n}=\left\{(z_{1},\dots ,z_{n})\mid z_{i}\in \mathbb {C} \right\}}$$

$${\displaystyle \mathbb {C} ^{n}=\underbrace {\mathbb {C} \times \mathbb {C} \times \cdots \times \mathbb {C} } _{n}.}$$

${\displaystyle z_{i}}$

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

See also

• Coordinate space

References

• Gunning, Robert; Hugo Rossi, Analytic functions of several complex variables