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

\mathbb C^n=\{ (z_1,\dots,z_n)|z_i\in\mathbb C\}

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

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 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.

References

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

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Complex coordinate space

See also

References