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 \Complex^n, and is the n-fold Cartesian product of the complex line \Complex with itself. Symbolically,
\Complex^n = \left\{ (z_1,\dots,z_n) \mid z_1, \dots, z_n \in \Complex\right\}
or
\Complex^n = \underbrace{\Complex \times \Complex \times \cdots \times \Complex}_{n}.
The variables z_i are the (complex) coordinates on the complex n-space.
The special case \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 \Complex^n with the 2n-dimensional real coordinate space, \mathbb R^{2n}. With the standard Euclidean topology, \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.
See also
References
- Gunning, Robert & Hugo Rossi, "Analytic functions of several complex variables"