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} \}

or

\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.

See also

References

This article is issued from Wikipedia - version of the Friday, December 11, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.