Holomorphic mapping

A mapping of a domain into a domain under which

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

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 " , Springer (1980) pp. Chapt. 15
[a2]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)