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''Riemann–Schwarz symmetry principle''
 
''Riemann–Schwarz symmetry principle''
  
 
A method of extending conformal mappings and analytic functions of a complex variable, formulated by B. Riemann and justified by H.A. Schwarz in the 19th century.
 
A method of extending conformal mappings and analytic functions of a complex variable, formulated by B. Riemann and justified by H.A. Schwarz in the 19th century.
  
The Riemann–Schwarz principle for conformal mappings is as follows. Let two domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819902.png" /> in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819903.png" /> be symmetric with respect to the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819904.png" />, let them be non-intersecting, and let their boundaries contain a common interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819905.png" />, whereby <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819906.png" /> is also a domain. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r0819909.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199010.png" /> be similarly defined. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199011.png" />, continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199012.png" />, conformally maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199013.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199014.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199015.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199016.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199017.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199018.png" /> and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199019.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199020.png" /> realizes a [[Conformal mapping|conformal mapping]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199022.png" />.
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The Riemann–Schwarz principle for conformal mappings is as follows. Let two domains $  D _ {1} $,  
 +
$  D _ {2} $
 +
in the complex plane $  \mathbf C $
 +
be symmetric with respect to the real axis $  \mathbf R $,  
 +
let them be non-intersecting, and let their boundaries contain a common interval $  \gamma \subset  \mathbf R $,  
 +
whereby $  D = D _ {1} \cup \gamma \cup D _ {2} $
 +
is also a domain. Let $  D _ {1}  ^  \star  $,  
 +
$  D _ {2}  ^  \star  $,  
 +
$  \gamma  ^  \star  $,  
 +
and $  D  ^  \star  $
 +
be similarly defined. If a function $  f _ {1} $,  
 +
continuous in $  D _ {1} \cup \gamma $,  
 +
conformally maps $  D _ {1} $
 +
onto $  D _ {1}  ^  \star  $
 +
and if $  f _ {1} ( \gamma ) = \gamma  ^  \star  $,  
 +
then the function $  f( z) $
 +
equal to $  f _ {1} ( z) $
 +
when $  z \in D _ {1} \cup \gamma $
 +
and to $  \overline{ {f _ {1} ( z) }}\; $
 +
when $  z \in D _ {2} $
 +
realizes a [[Conformal mapping|conformal mapping]] of $  D $
 +
onto $  D  ^  \star  $.
  
A more general formulation of the Riemann–Schwarz principle is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199026.png" /> are domains on the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199027.png" /> that are symmetric with respect to two neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199028.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199030.png" /> are open arcs, (see [[Symmetry principle|Symmetry principle]]).
+
A more general formulation of the Riemann–Schwarz principle is obtained when $  D _ {1} $,  
 +
$  D _ {2} $
 +
and $  D _ {1}  ^  \star  $,  
 +
$  D _ {2}  ^  \star  $
 +
are domains on the Riemann sphere $  \overline{\mathbf C}\; $
 +
that are symmetric with respect to two neighbourhoods $  C, C  ^  \star  \subset  \overline{\mathbf C}\; $,
 +
respectively, and $  \gamma \subset  C $,  
 +
$  \gamma \subset  C  ^  \star  $
 +
are open arcs, (see [[Symmetry principle|Symmetry principle]]).
  
The Riemann–Schwarz principle for holomorphic functions. Let the boundary of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199031.png" /> contain a real-analytic arc. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199032.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199033.png" />, continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199034.png" /> and if its values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199035.png" /> belong to another real-analytic arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199037.png" /> can be analytically extended to a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199038.png" />.
+
The Riemann–Schwarz principle for holomorphic functions. Let the boundary of a domain $  D \subset  \mathbf C $
 +
contain a real-analytic arc. If a function $  f $
 +
is holomorphic in $  D $,  
 +
continuous in $  D \cup \gamma $
 +
and if its values on $  \gamma $
 +
belong to another real-analytic arc $  \gamma  ^  \star  $,  
 +
then $  f $
 +
can be analytically extended to a neighbourhood of $  \gamma $.
  
 
The Riemann–Schwarz principle is used in the construction of conformal mappings of plane domains as well as in the theory of analytic extension of functions of one or several complex variables.
 
The Riemann–Schwarz principle is used in the construction of conformal mappings of plane domains as well as in the theory of analytic extension of functions of one or several complex variables.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Riemann–Schwarz principle is also known as the Schwarz reflection principle (cf. also [[Schwarz symmetry theorem|Schwarz symmetry theorem]]). The principle can be adapted to the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199039.png" />-arcs. Then one obtains a non-holomorphic extension. This can be used to prove smoothness up to the boundary of conformal mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199040.png" />-smoothly bounded domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199041.png" />. Moreover, this method has been generalized to obtain smoothness up to the boundary of biholomorphic mappings between strictly pseudo-convex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081990/r08199042.png" />-smoothly bounded domains, cf. [[#References|[a2]]], [[#References|[a3]]].
+
The Riemann–Schwarz principle is also known as the Schwarz reflection principle (cf. also [[Schwarz symmetry theorem|Schwarz symmetry theorem]]). The principle can be adapted to the case of $  C  ^ {k} $-arcs. Then one obtains a non-holomorphic extension. This can be used to prove smoothness up to the boundary of conformal mappings of $  C  ^ {k} $-smoothly bounded domains in $  \mathbf C $.  
 +
Moreover, this method has been generalized to obtain smoothness up to the boundary of biholomorphic mappings between strictly pseudo-convex, $  C  ^ {k} $-smoothly bounded domains, cf. [[#References|[a2]]], [[#References|[a3]]].
  
 
The analogue for holomorphic functions of the Schwarz reflection principle is the famous so-called edge-of-the-wedge theorem, [[#References|[a6]]].
 
The analogue for holomorphic functions of the Schwarz reflection principle is the famous so-called edge-of-the-wedge theorem, [[#References|[a6]]].

Latest revision as of 06:45, 13 June 2022


Riemann–Schwarz symmetry principle

A method of extending conformal mappings and analytic functions of a complex variable, formulated by B. Riemann and justified by H.A. Schwarz in the 19th century.

The Riemann–Schwarz principle for conformal mappings is as follows. Let two domains $ D _ {1} $, $ D _ {2} $ in the complex plane $ \mathbf C $ be symmetric with respect to the real axis $ \mathbf R $, let them be non-intersecting, and let their boundaries contain a common interval $ \gamma \subset \mathbf R $, whereby $ D = D _ {1} \cup \gamma \cup D _ {2} $ is also a domain. Let $ D _ {1} ^ \star $, $ D _ {2} ^ \star $, $ \gamma ^ \star $, and $ D ^ \star $ be similarly defined. If a function $ f _ {1} $, continuous in $ D _ {1} \cup \gamma $, conformally maps $ D _ {1} $ onto $ D _ {1} ^ \star $ and if $ f _ {1} ( \gamma ) = \gamma ^ \star $, then the function $ f( z) $ equal to $ f _ {1} ( z) $ when $ z \in D _ {1} \cup \gamma $ and to $ \overline{ {f _ {1} ( z) }}\; $ when $ z \in D _ {2} $ realizes a conformal mapping of $ D $ onto $ D ^ \star $.

A more general formulation of the Riemann–Schwarz principle is obtained when $ D _ {1} $, $ D _ {2} $ and $ D _ {1} ^ \star $, $ D _ {2} ^ \star $ are domains on the Riemann sphere $ \overline{\mathbf C}\; $ that are symmetric with respect to two neighbourhoods $ C, C ^ \star \subset \overline{\mathbf C}\; $, respectively, and $ \gamma \subset C $, $ \gamma \subset C ^ \star $ are open arcs, (see Symmetry principle).

The Riemann–Schwarz principle for holomorphic functions. Let the boundary of a domain $ D \subset \mathbf C $ contain a real-analytic arc. If a function $ f $ is holomorphic in $ D $, continuous in $ D \cup \gamma $ and if its values on $ \gamma $ belong to another real-analytic arc $ \gamma ^ \star $, then $ f $ can be analytically extended to a neighbourhood of $ \gamma $.

The Riemann–Schwarz principle is used in the construction of conformal mappings of plane domains as well as in the theory of analytic extension of functions of one or several complex variables.

References

[1] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)

Comments

The Riemann–Schwarz principle is also known as the Schwarz reflection principle (cf. also Schwarz symmetry theorem). The principle can be adapted to the case of $ C ^ {k} $-arcs. Then one obtains a non-holomorphic extension. This can be used to prove smoothness up to the boundary of conformal mappings of $ C ^ {k} $-smoothly bounded domains in $ \mathbf C $. Moreover, this method has been generalized to obtain smoothness up to the boundary of biholomorphic mappings between strictly pseudo-convex, $ C ^ {k} $-smoothly bounded domains, cf. [a2], [a3].

The analogue for holomorphic functions of the Schwarz reflection principle is the famous so-called edge-of-the-wedge theorem, [a6].

References

[a1] Z. Nehari, "Conformal mapping" , Dover, reprint (1975)
[a2] L. Nirenberg, S. Webster, P. Yang, "Local boundary regularity of holomorphic mappings" Comm. Pure Appl. Math. , 33 (1980) pp. 305–338
[a3] S.I. Pinchuk, S.V. Khasanov, "Asymptotically holomorphic functions and their applications" Math. USSR-Sb. , 62 : 2 (1989) pp. 541–550 Mat. Sb. , 134 (176) (1987) pp. 546–555; 576
[a4] C. Carathéodory, "Theory of functions" , 2 , Chelsea, reprint (1954)
[a5] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
[a6] W. Rudin, "Lectures on the edge-of-the-wedge theorem" , Amer. Math. Soc. (1971)
How to Cite This Entry:
Riemann-Schwarz principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann-Schwarz_principle&oldid=22985
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article