Holomorphic mapping
A mapping of a domain
into a domain
under which
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where all coordinate functions are holomorphic in
. If
, a holomorphic mapping coincides with a holomorphic function (cf. Analytic function).
A holomorphic mapping is called non-degenerate at a point if the rank of the Jacobian matrix
is maximal at
(and hence equals
). A holomorphic mapping is said to be non-degenerate in the domain
if it is non-degenerate at all points
. If
, the non-degeneracy of
is equivalent to the condition
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If , a non-degenerate holomorphic mapping is a conformal mapping. If
, a non-degenerate holomorphic mapping does not, in general, preserve angles between directions. If a holomorphic mapping
is non-degenerate at a point
and if
, then
is locally invertible, i.e., then there exist neighbourhoods
,
,
,
, and a holomorphic mapping
such that
for all
. If a holomorphic mapping
maps
onto
in a one-to-one correspondence and if
, then
is non-degenerate in
; if
, this is not true, e.g.
,
,
. If
and if
is non-degenerate in
, then the image of
is also a domain in
; if
, the principle of invariance of domain does not hold for mappings that are degenerate at certain points, e.g.
,
.
If and
are complex manifolds,
and
are atlases of their local coordinate systems (
,
are homeomorphisms; cf. Manifold), then a mapping
is said to be holomorphic if
is a holomorphic mapping for all
and
. 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 ![]() |
[a2] | S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) |
Holomorphic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphic_mapping&oldid=47245