# 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 $D _ {1}$, $D _ {2}$ in the complex plane $\mathbf C$ be symmetric with respect to the real axis $\mathbf R$, let them be non-intersecting, and let their boundaries contain a common interval $\gamma \subset \mathbf R$, whereby $D = D _ {1} \cup \gamma \cup D _ {2}$ is also a domain. Let $D _ {1} ^ \star$, $D _ {2} ^ \star$, $\gamma ^ \star$, and $D ^ \star$ be similarly defined. If a function $f _ {1}$, continuous in $D _ {1} \cup \gamma$, conformally maps $D _ {1}$ onto $D _ {1} ^ \star$ and if $f _ {1} ( \gamma ) = \gamma ^ \star$, then the function $f( z)$ equal to $f _ {1} ( z)$ when $z \in D _ {1} \cup \gamma$ and to $\overline{ {f _ {1} ( z) }}\;$ when $z \in D _ {2}$ realizes a conformal mapping of $D$ onto $D ^ \star$.

A more general formulation of the Riemann–Schwarz principle is obtained when $D _ {1}$, $D _ {2}$ and $D _ {1} ^ \star$, $D _ {2} ^ \star$ are domains on the Riemann sphere $\overline{\mathbf C}\;$ that are symmetric with respect to two neighbourhoods $C, C ^ \star \subset \overline{\mathbf C}\;$, respectively, and $\gamma \subset C$, $\gamma \subset C ^ \star$ are open arcs, (see Symmetry principle).

The Riemann–Schwarz principle for holomorphic functions. Let the boundary of a domain $D \subset \mathbf C$ contain a real-analytic arc. If a function $f$ is holomorphic in $D$, continuous in $D \cup \gamma$ and if its values on $\gamma$ belong to another real-analytic arc $\gamma ^ \star$, then $f$ can be analytically extended to a neighbourhood of $\gamma$.

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.

How to Cite This Entry:
Riemann–Schwarz principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann%E2%80%93Schwarz_principle&oldid=22986