Complex coordinate space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the <a href="/facts/Complex_dimension/e7M7pctQ">n-dimensional</a> complex coordinate space (or complex n-space) is the set of all ordered n-<a href="/facts/Tuple/BI1HIxL8">tuples</a> of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, also known as <a href="/facts/Complex_vector/M1pxMLx2">complex vectors</a>. The space is denoted 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
, and is the n-fold <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of the <a href="/facts/Complex_line/muAoqeqx">complex line</a> 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 with itself. Symbolically,

C
          
          
            n
          
        
        =
        
          {
          
            (
            
              z
              
                1
              
            
            ,
            …
            ,
            
              z
              
                n
              
            
            )
            ∣
            
              z
              
                i
              
            
            ∈
            
              C
            
          
          }
        
      
    
    {\displaystyle \mathbb {C} ^{n}=\left\{(z_{1},\dots ,z_{n})\mid z_{i}\in \mathbb {C} \right\}}

C
          
          
            n
          
        
        =
        
          
            
              
                
                  C
                
                ×
                
                  C
                
                ×
                ⋯
                ×
                
                  C
                
              
              ⏟
            
          
          
            n
          
        
        .
      
    
    {\displaystyle \mathbb {C} ^{n}=\underbrace {\mathbb {C} \times \mathbb {C} \times \cdots \times \mathbb {C} } _{n}.}

The variables 
 
 
 
 
 z
 
 i
 
 
 
 
 {\displaystyle z_{i}}
 
 are the (complex) coordinates on the complex n-space. 
The special case 
 
 
 
 
 
 C
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {C} ^{2}}
 
, called the complex coordinate plane, is not to be confused with the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>, a graphical representation of the complex line.
Complex coordinate space is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over the complex numbers, with componentwise addition and <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a>. The real and imaginary parts of the coordinates set up a <a href="/facts/Bijection/j4gTuTmW">bijection</a> of 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
 with the <a href="/facts/Dimension/0ktmUyac">2n-dimensional</a> <a href="/facts/Real_coordinate_space/tEtvw03F">real coordinate space</a>, 
 
 
 
 
 
 R
 
 
 2
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{2n}}
 
. With the standard <a href="/facts/Euclidean_topology/gCcjWJvA">Euclidean topology</a>, 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
 is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> over the complex numbers.
A function on an open subset of complex n-space is holomorphic if it is <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> in each complex coordinate separately. <a href="/facts/Several_complex_variables/axwNpV4a">Several complex variables</a> 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 <a href="/facts/Complex_manifold/DlvPnpzm">complex manifolds</a>.

Complex coordinate space open-in-new

Complex coordinate space