Riemann theorem

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 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 $ w = f( z) $, which guarantee its uniqueness, Riemann used for the same purpose the conditions "fa= b, fz=w" , 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, 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 $ 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).

L.D. Kudryavtsev

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.