mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-

tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

s of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. It is denoted

\Complex^n

, and is the ''n''-fold

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

of the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

\Complex

with itself. Symbolically,

\Complex^n = \left\

\Complex^n = \underbrace_.

The variables

z_i

are the (complex) coordinates on the complex ''n''-space. Complex coordinate space is a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

over the complex numbers, with componentwise addition and scalar multiplication. The real and imaginary parts of the coordinates set up a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

\Complex^n

with the 2''n''-dimensional

real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...

\mathbb R^

. With the standard

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...

\Complex^n

is a

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

over the complex numbers. A function on an open subset of complex ''n''-space is holomorphic if it is

holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

in each complex coordinate separately.

Several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...

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 manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

References

* {{citation, author1-link = Robert Gunning (mathematician), first = Robert , last = Gunning, author2=Hugo Rossi , title=Analytic functions of several complex variables Several complex variables Topological vector spaces

See also

References