Namespaces
Variants
Actions

Riemann theorem

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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 "$f(a)= b, \mathop{\rm arg} f^\prime(a)=\alpha$" 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, 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.

How to Cite This Entry:
Riemann theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_theorem&oldid=53719
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article