# Riemann-Schwarz principle

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Riemann–Schwarz symmetry principle

A method of extending conformal mappings and analytic functions of a complex variable, formulated by B. Riemann and justified by H.A. Schwarz in the 19th century.

The Riemann–Schwarz principle for conformal mappings is as follows. Let two domains , in the complex plane be symmetric with respect to the real axis , let them be non-intersecting, and let their boundaries contain a common interval , whereby is also a domain. Let , , , and be similarly defined. If a function , continuous in , conformally maps onto and if , then the function equal to when and to when realizes a conformal mapping of onto .

A more general formulation of the Riemann–Schwarz principle is obtained when , and , are domains on the Riemann sphere that are symmetric with respect to two neighbourhoods , respectively, and , are open arcs, (see Symmetry principle).

The Riemann–Schwarz principle for holomorphic functions. Let the boundary of a domain contain a real-analytic arc. If a function is holomorphic in , continuous in and if its values on belong to another real-analytic arc , then can be analytically extended to a neighbourhood of .

The Riemann–Schwarz principle is used in the construction of conformal mappings of plane domains as well as in the theory of analytic extension of functions of one or several complex variables.

#### References

 [1] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)