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 , and is the n-fold Cartesian product of the complex plane with itself. Symbolically,
The variables 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 with the real coordinate space . With the standard Euclidean topology, 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.
- Robert Gunning; Hugo Rossi, Analytic functions of several complex variables