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Complex coordinate space

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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 , and is the n-fold Cartesian product of the complex line with itself. Symbolically, or The variables are the (complex) coordinates on the complex n-space. The special case , 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 with the 2n-dimensional 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.

See also

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References

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