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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475601.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475602.png" /> into a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475603.png" /> under which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475604.png" /></td> </tr></table>
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where all coordinate functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475605.png" /> are holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475606.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475607.png" />, a holomorphic mapping coincides with a holomorphic function (cf. [[Analytic function|Analytic function]]).
+
A mapping  $  f: D \rightarrow D ^ { \prime } $
 +
of a domain  $  D \subset  \mathbf C  ^ {n} $
 +
into a domain  $  D ^ { \prime } \subset  \mathbf C  ^ {m} $
 +
under which
  
A holomorphic mapping is called non-degenerate at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475608.png" /> if the rank of the Jacobian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h0475609.png" /> is maximal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756010.png" /> (and hence equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756011.png" />). A holomorphic mapping is said to be non-degenerate in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756012.png" /> if it is non-degenerate at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756014.png" />, the non-degeneracy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756015.png" /> is equivalent to the condition
+
$$
 +
z = ( z _ {1} \dots z _ {n} )  \rightarrow \
 +
( f _ {1} ( z) \dots f _ {m} ( z)),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756016.png" /></td> </tr></table>
+
where all coordinate functions  $  f _ {1} \dots f _ {m} $
 +
are holomorphic in  $  D $.
 +
If  $  m = 1 $,
 +
a holomorphic mapping coincides with a holomorphic function (cf. [[Analytic function|Analytic function]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756017.png" />, a non-degenerate holomorphic mapping is a conformal mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756018.png" />, a non-degenerate holomorphic mapping does not, in general, preserve angles between directions. If a holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756019.png" /> is non-degenerate at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756020.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756022.png" /> is locally invertible, i.e., then there exist neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756026.png" />, and a holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756029.png" />. If a holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756030.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756031.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756032.png" /> in a one-to-one correspondence and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756034.png" /> is non-degenerate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756035.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756036.png" />, this is not true, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756040.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756041.png" /> is non-degenerate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756042.png" />, then the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756043.png" /> is also a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756044.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756045.png" />, the principle of invariance of domain does not hold for mappings that are degenerate at certain points, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756047.png" />.
+
A holomorphic mapping is called non-degenerate at a point $  z \in D $
 +
if the rank of the Jacobian matrix  $  \| \partial  f / \partial  z \| $
 +
is maximal at  $  z $(
 +
and hence equals  $  \min ( n, m) $).  
 +
A holomorphic mapping is said to be non-degenerate in the domain  $  D $
 +
if it is non-degenerate at all points  $  z \in D $.
 +
If  $  m = n $,  
 +
the non-degeneracy of $  f $
 +
is equivalent to the condition
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756049.png" /> are complex manifolds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756051.png" /> are atlases of their local coordinate systems (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756053.png" /> are homeomorphisms; cf. [[Manifold|Manifold]]), then a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756054.png" /> is said to be holomorphic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756055.png" /> is a holomorphic mapping for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047560/h04756057.png" />. Holomorphic mappings of complex spaces are defined in a similar manner (cf. [[Analytic mapping|Analytic mapping]]). See also [[Biholomorphic mapping|Biholomorphic mapping]].
+
$$
 +
\mathop{\rm det}  \left \|
 +
\frac{\partial  f }{\partial  z }
 +
\
 +
\right \|  \neq  0.
 +
$$
 +
 
 +
If $  n = m = 1 $,
 +
a non-degenerate holomorphic mapping is a conformal mapping. If  $  n = m \geq  2 $,
 +
a non-degenerate holomorphic mapping does not, in general, preserve angles between directions. If a holomorphic mapping  $  f $
 +
is non-degenerate at a point  $  a \in D $
 +
and if  $  m = n $,
 +
then  $  f $
 +
is locally invertible, i.e., then there exist neighbourhoods  $  U $,
 +
$  U ^ { \prime } $,
 +
$  a \in U \subset  D $,
 +
$  f( a) \in U ^ { \prime } \subset  D ^ { \prime } $,
 +
and a holomorphic mapping  $  f ^ { - 1 } :  U ^ { \prime } \rightarrow U $
 +
such that  $  f ^ { - 1 } \circ f( z) = z $
 +
for all  $  z \in U $.
 +
If a holomorphic mapping  $  f $
 +
maps  $  D $
 +
onto  $  f( D) $
 +
in a one-to-one correspondence and if  $  m = n $,
 +
then  $  f $
 +
is non-degenerate in  $  D $;
 +
if  $  m > n $,
 +
this is not true, e.g. $  z \rightarrow ( z  ^ {2} , z  ^ {3} ) $,
 +
$  D = \mathbf C $,
 +
$  D ^ { \prime } = \mathbf C  ^ {2} $.  
 +
If  $  m \leq  n $
 +
and if  $  f $
 +
is non-degenerate in  $  D $,
 +
then the image of  $  D $
 +
is also a domain in  $  \mathbf C  ^ {m} $;
 +
if  $  m > 1 $,
 +
the principle of invariance of domain does not hold for mappings that are degenerate at certain points, e.g. $  ( z _ {1} , z _ {2} ) \rightarrow ( z _ {1} , z _ {1} z _ {2} ) $,
 +
$  D = D ^ { \prime } = \mathbf C  ^ {2} $.
 +
 
 +
If  $  M $
 +
and  $  M ^ { \prime } $
 +
are complex manifolds,  $  \{ ( U _  \alpha  , \phi _  \alpha  ) \} $
 +
and  $  \{ ( U _  \beta  ^ { \prime } , \phi _  \beta  ^  \prime  ) \} $
 +
are atlases of their local coordinate systems ( $  \phi _  \alpha  : U _  \alpha  \rightarrow D _  \alpha  \subset  \mathbf C  ^ {n} $,  
 +
$  \phi _  \beta  ^  \prime  : U _  \beta  ^ { \prime } \rightarrow D _  \beta  ^ { \prime } \subset  \mathbf C  ^ {m} $
 +
are homeomorphisms; cf. [[Manifold|Manifold]]), then a mapping $  f: M \rightarrow M ^ { \prime } $
 +
is said to be holomorphic if $  \phi _  \beta  ^  \prime  \circ f \circ \phi _  \alpha  ^ {-} 1 : D _  \alpha  \rightarrow D _  \beta  ^ { \prime } $
 +
is a holomorphic mapping for all $  \alpha $
 +
and $  \beta $.  
 +
Holomorphic mappings of complex spaces are defined in a similar manner (cf. [[Analytic mapping|Analytic mapping]]). See also [[Biholomorphic mapping|Biholomorphic mapping]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 22:10, 5 June 2020


A mapping $ f: D \rightarrow D ^ { \prime } $ of a domain $ D \subset \mathbf C ^ {n} $ into a domain $ D ^ { \prime } \subset \mathbf C ^ {m} $ under which

$$ z = ( z _ {1} \dots z _ {n} ) \rightarrow \ ( f _ {1} ( z) \dots f _ {m} ( z)), $$

where all coordinate functions $ f _ {1} \dots f _ {m} $ are holomorphic in $ D $. If $ m = 1 $, a holomorphic mapping coincides with a holomorphic function (cf. Analytic function).

A holomorphic mapping is called non-degenerate at a point $ z \in D $ if the rank of the Jacobian matrix $ \| \partial f / \partial z \| $ is maximal at $ z $( and hence equals $ \min ( n, m) $). A holomorphic mapping is said to be non-degenerate in the domain $ D $ if it is non-degenerate at all points $ z \in D $. If $ m = n $, the non-degeneracy of $ f $ is equivalent to the condition

$$ \mathop{\rm det} \left \| \frac{\partial f }{\partial z } \ \right \| \neq 0. $$

If $ n = m = 1 $, a non-degenerate holomorphic mapping is a conformal mapping. If $ n = m \geq 2 $, a non-degenerate holomorphic mapping does not, in general, preserve angles between directions. If a holomorphic mapping $ f $ is non-degenerate at a point $ a \in D $ and if $ m = n $, then $ f $ is locally invertible, i.e., then there exist neighbourhoods $ U $, $ U ^ { \prime } $, $ a \in U \subset D $, $ f( a) \in U ^ { \prime } \subset D ^ { \prime } $, and a holomorphic mapping $ f ^ { - 1 } : U ^ { \prime } \rightarrow U $ such that $ f ^ { - 1 } \circ f( z) = z $ for all $ z \in U $. If a holomorphic mapping $ f $ maps $ D $ onto $ f( D) $ in a one-to-one correspondence and if $ m = n $, then $ f $ is non-degenerate in $ D $; if $ m > n $, this is not true, e.g. $ z \rightarrow ( z ^ {2} , z ^ {3} ) $, $ D = \mathbf C $, $ D ^ { \prime } = \mathbf C ^ {2} $. If $ m \leq n $ and if $ f $ is non-degenerate in $ D $, then the image of $ D $ is also a domain in $ \mathbf C ^ {m} $; if $ m > 1 $, the principle of invariance of domain does not hold for mappings that are degenerate at certain points, e.g. $ ( z _ {1} , z _ {2} ) \rightarrow ( z _ {1} , z _ {1} z _ {2} ) $, $ D = D ^ { \prime } = \mathbf C ^ {2} $.

If $ M $ and $ M ^ { \prime } $ are complex manifolds, $ \{ ( U _ \alpha , \phi _ \alpha ) \} $ and $ \{ ( U _ \beta ^ { \prime } , \phi _ \beta ^ \prime ) \} $ are atlases of their local coordinate systems ( $ \phi _ \alpha : U _ \alpha \rightarrow D _ \alpha \subset \mathbf C ^ {n} $, $ \phi _ \beta ^ \prime : U _ \beta ^ { \prime } \rightarrow D _ \beta ^ { \prime } \subset \mathbf C ^ {m} $ are homeomorphisms; cf. Manifold), then a mapping $ f: M \rightarrow M ^ { \prime } $ is said to be holomorphic if $ \phi _ \beta ^ \prime \circ f \circ \phi _ \alpha ^ {-} 1 : D _ \alpha \rightarrow D _ \beta ^ { \prime } $ is a holomorphic mapping for all $ \alpha $ and $ \beta $. Holomorphic mappings of complex spaces are defined in a similar manner (cf. Analytic mapping). See also Biholomorphic mapping.

References

[1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)

Comments

A non-degenerate mapping is also called non-singular.

References

[a1] W. Rudin, "Function theory in the unit ball in " , Springer (1980) pp. Chapt. 15
[a2] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
How to Cite This Entry:
Holomorphic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphic_mapping&oldid=47245
This article was adapted from an original article by E.D. SolomentsevE.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article