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Riemann's theorem on conformal mappings: Given any two simply-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820802.png" /> of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820803.png" />, distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820804.png" /> and also from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820805.png" /> with a point excluded from it, then an infinite number of analytic single-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820806.png" /> can be found such that each one realizes a one-to-one conformal transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820807.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820808.png" />. In this case, for any pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208011.png" /> and any real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208013.png" />, a unique function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208014.png" /> of this class can be found for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208016.png" />. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208017.png" /> geometrically means that each infinitely-small vector emanating from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208018.png" /> changes under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208019.png" /> into an infinitely-small vector the direction of which forms with the direction of the original vector the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208020.png" />.
 
Riemann's theorem on conformal mappings: Given any two simply-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820802.png" /> of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820803.png" />, distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820804.png" /> and also from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820805.png" /> with a point excluded from it, then an infinite number of analytic single-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820806.png" /> can be found such that each one realizes a one-to-one conformal transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820807.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820808.png" />. In this case, for any pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r0820809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208011.png" /> and any real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208013.png" />, a unique function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208014.png" /> of this class can be found for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208016.png" />. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208017.png" /> geometrically means that each infinitely-small vector emanating from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208018.png" /> changes under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208019.png" /> into an infinitely-small vector the direction of which forms with the direction of the original vector the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208020.png" />.
  
Riemann's theorem is fundamental in the theory of [[Conformal mapping|conformal mapping]] and in the geometrical theory of functions of a complex variable in general. In addition to its generalizations to multiply-connected domains, it finds wide application in the theory of functions of a complex variable, in mathematical physics, in the theory of elasticity, in aero- and hydromechanics, in electro- and magnetostatics, etc. This theorem was formulated by B. Riemann (1851) for the more general case of simply-connected and, generally speaking, non-single sheeted domains over the complex plane. Instead of using the normalizing conditions "fa= b, argf'a=a" of the conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208021.png" />, which guarantee its uniqueness, Riemann used for the same purpose the conditions "fa= b, fz=w" , where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208025.png" /> are points of the boundaries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208027.png" />, respectively, given in advance. The last conditions are not always correct, given the contemporary definition of a simply-connected domain. In proving his theorem, Riemann drew to a considerable degree on concepts of physics, which also convinced him of the importance of this theorem for applications. D. Hilbert made Riemann's proof mathematically precise by substantiating the so-called [[Dirichlet principle|Dirichlet principle]], which was used by Riemann in his proof.
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Riemann's theorem is fundamental in the theory of [[Conformal mapping|conformal mapping]] and in the geometrical theory of functions of a complex variable in general. In addition to its generalizations to multiply-connected domains, it finds wide application in the theory of functions of a complex variable, in mathematical physics, in the theory of elasticity, in aero- and hydromechanics, in electro- and magnetostatics, etc. This theorem was formulated by B. Riemann (1851) for the more general case of simply-connected and, generally speaking, non-single sheeted domains over the complex plane. Instead of using the normalizing conditions "fa= b, argf'a=a" of the conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208021.png" />, which guarantee its uniqueness, Riemann used for the same purpose the conditions "fa= b, fz=w" , where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208025.png" /> are points of the boundaries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208027.png" />, respectively, given in advance. The last conditions are not always correct, given the contemporary definition of a simply-connected domain. In proving his theorem, Riemann drew to a considerable degree on concepts of physics, which also convinced him of the importance of this theorem for applications. D. Hilbert made Riemann's proof mathematically precise by substantiating the so-called [[Dirichlet principle|Dirichlet principle]], which was used by Riemann in his proof.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann,   "Gesammelte mathematischen Abhandlungen" , Dover, reprint (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow,   "Einführung in die Funktionentheorie" , '''1–3''' , Teubner (1958–1959) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Gesammelte mathematischen Abhandlungen" , Dover, reprint (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , '''1–3''' , Teubner (1958–1959) (Translated from Russian) {{MR|0342680}} {{MR|0264037}} {{MR|0264036}} {{MR|0264038}} {{MR|0123686}} {{MR|0123685}} {{MR|0098843}} {{ZBL|0177.33401}} {{ZBL|0141.26003}} {{ZBL|0141.26002}} {{ZBL|0082.28802}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Translated from Russian) {{MR|0247039}} {{ZBL|0183.07502}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Z. Nehari,   "Conformal mapping" , Dover, reprint (1975)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Z. Nehari, "Conformal mapping" , Dover, reprint (1975) {{MR|0377031}} {{ZBL|0071.07301}} {{ZBL|0052.08201}} {{ZBL|0048.31503}} {{ZBL|0041.41201}} </TD></TR></table>
  
 
Riemann's theorem on the rearrangement of terms of a series: If a series in which the terms are real numbers converges but does not converge absolutely, then for any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208028.png" /> there is a rearrangement of the terms of this series such that the sum of the series obtained will be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208029.png" />. Furthermore, there is a rearrangement of the terms of the series such that its sum will be equal to one of the previously given signed infinities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208030.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208031.png" />, and also such that its sum will not be equal either to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208032.png" /> or to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208033.png" />, but the sequences of its partial sums have given liminf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208034.png" /> and limsup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208035.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208036.png" /> (see [[Series|Series]]).
 
Riemann's theorem on the rearrangement of terms of a series: If a series in which the terms are real numbers converges but does not converge absolutely, then for any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208028.png" /> there is a rearrangement of the terms of this series such that the sum of the series obtained will be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208029.png" />. Furthermore, there is a rearrangement of the terms of the series such that its sum will be equal to one of the previously given signed infinities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208030.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208031.png" />, and also such that its sum will not be equal either to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208032.png" /> or to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208033.png" />, but the sequences of its partial sums have given liminf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208034.png" /> and limsup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208035.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082080/r08208036.png" /> (see [[Series|Series]]).
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Knopp,   "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 &amp; Dover, reprint, 1990) {{MR|0028430}} {{ZBL|0124.28302}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 {{MR|0385023}} {{ZBL|0346.26002}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
Another "Riemann theorem" is the Riemann removable singularities theorem, see [[Removable set|Removable set]].
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Another "Riemann theorem" is the Riemann removable singularities theorem, see [[Removable set|Removable set]].

Revision as of 17:01, 15 April 2012

Riemann's theorem on conformal mappings: Given any two simply-connected domains and of the extended complex plane , distinct from and also from with a point excluded from it, then an infinite number of analytic single-valued functions on can be found such that each one realizes a one-to-one conformal transformation of onto . In this case, for any pair of points , , and and any real number , , a unique function of this class can be found for which , . The condition geometrically means that each infinitely-small vector emanating from the point changes under the transformation into an infinitely-small vector the direction of which forms with the direction of the original vector the angle .

Riemann's theorem is fundamental in the theory of conformal mapping and in the geometrical theory of functions of a complex variable in general. In addition to its generalizations to multiply-connected domains, it finds wide application in the theory of functions of a complex variable, in mathematical physics, in the theory of elasticity, in aero- and hydromechanics, in electro- and magnetostatics, etc. This theorem was formulated by B. Riemann (1851) for the more general case of simply-connected and, generally speaking, non-single sheeted domains over the complex plane. Instead of using the normalizing conditions "fa= b, argf'a=a" of the conformal mapping , which guarantee its uniqueness, Riemann used for the same purpose the conditions "fa= b, fz=w" , where , and and are points of the boundaries of and , respectively, given in advance. The last conditions are not always correct, given the contemporary definition of a simply-connected domain. In proving his theorem, Riemann drew to a considerable degree on concepts of physics, which also convinced him of the importance of this theorem for applications. D. Hilbert made Riemann's proof mathematically precise by substantiating the so-called Dirichlet principle, which was used by Riemann in his proof.

References

[1] B. Riemann, "Gesammelte mathematischen Abhandlungen" , Dover, reprint (1953)
[2] I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian) MR0342680 MR0264037 MR0264036 MR0264038 MR0123686 MR0123685 MR0098843 Zbl 0177.33401 Zbl 0141.26003 Zbl 0141.26002 Zbl 0082.28802
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) MR0247039 Zbl 0183.07502


Comments

This theorem is also called the Riemann mapping theorem.

References

[a1] Z. Nehari, "Conformal mapping" , Dover, reprint (1975) MR0377031 Zbl 0071.07301 Zbl 0052.08201 Zbl 0048.31503 Zbl 0041.41201

Riemann's theorem on the rearrangement of terms of a series: If a series in which the terms are real numbers converges but does not converge absolutely, then for any number there is a rearrangement of the terms of this series such that the sum of the series obtained will be equal to . Furthermore, there is a rearrangement of the terms of the series such that its sum will be equal to one of the previously given signed infinities or , and also such that its sum will not be equal either to or to , but the sequences of its partial sums have given liminf and limsup , with (see Series).

L.D. Kudryavtsev

Comments

References

[a1] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) MR0028430 Zbl 0124.28302
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 MR0385023 Zbl 0346.26002

Comments

Another "Riemann theorem" is the Riemann removable singularities theorem, see Removable set.

How to Cite This Entry:
Riemann theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_theorem&oldid=11219
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article