Complex coordinate space

Inmathematics,then-dimensionalcomplex coordinate space(orcomplexn-space) is the set of all orderedn-tuplesofcomplex numbers,also known ascomplex vectors.The space is denoted$\mathbb {C} ^{n}$,and is then-foldCartesian productof thecomplex plane$\mathbb {C}$with itself. Symbolically, $$\mathbb {C} ^{n}=\left\{(z_{1},\dots,z_{n})\mid z_{i}\in \mathbb {C} \right\}$$ 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 complexn-space.

Complex coordinate space is avector spaceover the complex numbers, with componentwise addition andscalar multiplication.The real and imaginary parts of the coordinates set up abijectionof$\mathbb {C} ^{n}$with the 2n-dimensionalreal coordinate space,$\mathbb {R} ^{2n}$.With the standardEuclidean topology,$\mathbb {C} ^{n}$is atopological vector spaceover the complex numbers.

A function on an open subset of complexn-space is holomorphic if it isholomorphicin each complex coordinate separately.Several complex variablesis the study of such holomorphic functions innvariables. More generally, the complexn-space is the target space for holomorphic coordinate systems oncomplex manifolds.

See also

• Coordinate space

References

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