Symmetry principle

Schwarz' symmetry principle, Riemann–Schwarz symmetry principle, for analytic functions

Let a domain $ G $ in the extended complex plane $ \overline{\mathbf C} $ be bounded by a closed Jordan curve $ \Gamma $, part of which is an arc $ l $ of a circle $ L $ in $ \overline{\mathbf C} $. Further, let $ f ( z) $ be a function defined and continuous on $ G \cup l $, analytic in $ G $, and on $ l $ take values belonging to some circle $ C $ in $ \overline{\mathbf C} $. Then $ f ( z) $ can be extended across the arc $ l $ into the domain $ G ^ {*} $ that is symmetric with $ G $ relative to $ L $, to a function analytic in $ G \cup l \cup G ^ {*} $. Such an extension (across $ l $) is unique and is defined by the following property of the original function $ f ( z) $: If $ z \in G $ and $ z ^ {*} \in G ^ {*} $ are symmetric (inverse) relative to $ L $, then $ w = f ( z) $ and $ w ^ {*} = f ( z ^ {*} ) $ are symmetric relative to $ C $. In particular, if $ L $ and $ C $ coincide with the real axis in $ \overline{\mathbf C} $, then $ f ( z) = \overline{ {f ( \overline{z} ) }} $ for $ z \in G \cup l \cup G ^ {*} $. By circles in the extended complex plane one understands both proper circles and lines. Continuity also can be taken as usual and in a generalized sense, that is, $ f ( z) $ is called continuous at $ z _ {0} $ if $ f ( z) \rightarrow f ( z _ {0} ) $ as $ z \rightarrow z _ {0} $, independently of the finiteness or infiniteness of $ f ( z _ {0} ) $. The curve $ \Gamma $, as well as $ l $, may pass through the point at $ \infty $. From the conditions, $ f ( l) \subset C $, but it is not necessary that $ f ( l) = C $. In addition, if $ G $ and $ G ^ {*} $ have a common interior point, then the continued function need not be single-valued at these points.

The symmetry principle for harmonic functions for the same $ G $, $ L $, $ l $, $ G ^ {*} $ is formulated as follows: If a function $ u ( x, y) $ is harmonic in $ G $, continuous on $ G \cup l $ and equal to zero on $ l $, then $ u $ can be extended across $ l $ into $ G ^ {*} $ to a function that is harmonic in $ G \cup l \cup G ^ {*} $. Here, if $ ( x, y) \in G $ and $ ( x ^ {*} , y ^ {*} ) \in G ^ {*} $ are symmetric relative to $ L $, then $ u ( x ^ {*} , y ^ {*} ) = - u ( x, y) $.

The generalization of the symmetry principle to the case of an analytic arc $ l $( and $ C $) is the Schwarz principle of analytic continuation of analytic and harmonic functions (see [1], [2]). The generalization of the symmetry principle for harmonic functions to the case of a function of any number of variables is called the reflection principle. The symmetry principle is widely used in applications of the theory of analytic and harmonic functions (under conformal mappings of domains with one or more axes of symmetry, in the theories of elasticity, hydromechanics, electrostatics, etc.).

References

[1]	H.A. Schwarz, "Gesamm. math. Abhandl." , 2 , Springer (1890)
[2]	I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian)
[3]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)

Comments

The symmetry principle has interesting generalizations in the theory of holomorphic functions and mappings of several complex variables.

Examples are the edge-of-the-wedge theorem (see Bogolyubov theorem) and reflection principles for holomorphic mappings, which lead in many cases to smooth extendibility of such mappings to the boundary of the domains involved. See also Biholomorphic mapping.

References

[a1]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986)
[a2]	C. Carathéodory, "Theory of functions" , 2 , Chelsea, reprint (1981)
[a3]	E. Hille, "Analytic function theory" , 2 , Ginn (1959)