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. It is denoted ${\mathbb{ℂ}}^n$, and is the n-fold Cartesian product of the complex plane $\mathbb{ℂ}$ with itself. Symbolically, $${\mathbb{ℂ}}^n=\{(z_1,\dots ,z_n)\mid z_i\in \mathbb{ℂ}\}$$ or $${\mathbb{ℂ}}^n=\underset{n}{\underbrace{\mathbb{ℂ}\times \mathbb{ℂ}\times \cdots \times \mathbb{ℂ}}}.$$ The variables $z_i$ 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 ${\mathbb{ℂ}}^n$ with the 2n-dimensional real coordinate space, ${\mathbb{ℝ}}^{2n}$. With the standard Euclidean topology, ${\mathbb{ℂ}}^n$ 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