View source for Riemann theorem

<!--
r0820801.png
$#A+1 = 36 n = 0
$#C+1 = 36 : ~/encyclopedia/old_files/data/R082/R.0802080 Riemann theorem
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

==Riemann's theorem on conformal mappings==
Given any two simply-connected domains  $  G _ {1} $
and  $  G _ {2} $
of the extended complex plane  $  \overline{\mathbf C} $, 
distinct from  $  \overline{\mathbf C} $
and also from  $  \overline{\mathbf C} $
with a point excluded from it, then an infinite number of analytic single-valued functions on  $  G _ {1} $
can be found such that each one realizes a one-to-one conformal transformation of  $  G _ {1} $
onto  $  G _ {2} $. 
In this case, for any pair of points  $  a \in G _ {1} $, 
$  a \neq \infty $, 
and  $  b \in G _ {2} $
and any real number  $  \alpha $, 
$  0 \leq  \alpha \leq  2 \pi $, 
a unique function  $  f $
of this class can be found for which  $  f( a) = b $, 
$   \mathop{\rm arg}  f ^ { \prime } ( a) = \alpha $. 
The condition  $   \mathop{\rm arg}  f ^ { \prime } ( a) = \alpha $
geometrically means that each infinitely-small vector emanating from the point  $  a $
changes under the transformation  $  w = f( z) $
into an infinitely-small vector the direction of which forms with the direction of the original vector the angle  $  \alpha $.

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 "$f(a)= b, \mathop{\rm arg} f^\prime(a)=b$" of the conformal mapping  $  w = f( z) $, 
which guarantee its uniqueness, Riemann used for the same purpose the conditions "$f(a)= b, f(\zeta)=\omega$" , where  $  a \in G _ {1} $, $  b \in G _ {2} $ and  $  \zeta $ and  $  \omega $ are points of the boundaries of  $  G _ {1} $ and  $  G _ {2} $, 
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====
<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>

====Comments====
This theorem is also called the Riemann mapping theorem.

====References====
<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  $  A $
there is a rearrangement of the terms of this series such that the sum of the series obtained will be equal to  $  A $. 
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  $  + \infty $
or  $  - \infty $, 
and also such that its sum will not be equal either to  $  + \infty $
or to  $  - \infty $, 
but the sequences of its partial sums have given liminf  $  \lambda $
and limsup  $  \mu $, 
with  $  - \infty \leq  \lambda < \mu \leq  \infty $(
see [[Series|Series]]).

''L.D. Kudryavtsev''

====Comments====

====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) {{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====
Another "Riemann theorem" is the Riemann removable singularities theorem, see [[Removable set|Removable set]].