Difference between revisions of "Multi-sheeted region"
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| + | A region $ S $ | ||
| + | of a [[Riemann surface|Riemann surface]] $ R $, | ||
| + | considered as a covering surface over the complex plane $ \mathbf C $, | ||
| + | such that above each point of its projection $ D \subset \mathbf C $ | ||
| + | there are at least two points of $ S $; | ||
| + | a [[Branch point|branch point]] of $ R $ | ||
| + | of order $ k- 1 $ | ||
| + | is regarded here as $ k $ | ||
| + | distinct points. For example, the analytic function $ w = z ^ {2} $ | ||
| + | is a one-to-one mapping of the disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $ | ||
| + | onto the two-sheeted region (two-sheeted disc) $ S = \{ {w \in R } : {| w | < 1 } \} $ | ||
| + | of the Riemann surface $ R $ | ||
| + | of this function; this mapping is conformal everywhere except at the origin. | ||
| + | For analytic functions of several complex variables there arise multi-sheeted Riemann domains (cf. [[Riemannian domain|Riemannian domain]]) over the complex space $ \mathbf C ^ {n} $. | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.L. Siegel, "Topics in complex functions" , '''1''' , Wiley (Interscience) (1988) pp. Chapt. 1, Sect. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.L. Siegel, "Topics in complex functions" , '''1''' , Wiley (Interscience) (1988) pp. Chapt. 1, Sect. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10</TD></TR></table> | ||
Latest revision as of 08:01, 6 June 2020
A region $ S $
of a Riemann surface $ R $,
considered as a covering surface over the complex plane $ \mathbf C $,
such that above each point of its projection $ D \subset \mathbf C $
there are at least two points of $ S $;
a branch point of $ R $
of order $ k- 1 $
is regarded here as $ k $
distinct points. For example, the analytic function $ w = z ^ {2} $
is a one-to-one mapping of the disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
onto the two-sheeted region (two-sheeted disc) $ S = \{ {w \in R } : {| w | < 1 } \} $
of the Riemann surface $ R $
of this function; this mapping is conformal everywhere except at the origin.
For analytic functions of several complex variables there arise multi-sheeted Riemann domains (cf. Riemannian domain) over the complex space $ \mathbf C ^ {n} $.
Comments
References
| [a1] | C.L. Siegel, "Topics in complex functions" , 1 , Wiley (Interscience) (1988) pp. Chapt. 1, Sect. 4 |
| [a2] | G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 |
How to Cite This Entry:
Multi-sheeted region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-sheeted_region&oldid=47922
Multi-sheeted region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-sheeted_region&oldid=47922
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article