Schwarz reflection principle

For a region G define G*:={z:z¯G} (where z¯ is the complex conjugateMathworldPlanetmath of z). If G is a symmetric region, that is G=G*, then we define G+:={zG:Imz>0}, G-:={zG:Imz<0} and G0:={zG:Imz=0}.


Let GC be a region such that G=G* and suppose that f:G+G0C is a continuous functionsMathworldPlanetmathPlanetmath that is analyticPlanetmathPlanetmath on G+ and further that f(x) is real for xG0 (that is for real x), then there is an analytic function g:GC such that g(z)=f(z) for zG+G0.

That is you can “reflect” an analytic function across the real axis. Note that by composing with various conformal mappingsMathworldPlanetmathPlanetmath you could generalize the above to reflection across an analytic curve. So loosely stated, the theorem says that if an analytic function is defined in a region with some “nice” boundary and the function behaves “nice” on this boundary, then we can extend the function to a larger domain. Let us make this statement precise with the following generalizationPlanetmathPlanetmath.


Let G,ΩC be regions and let γ and ω be free analytic boundary arcs in G and Ω. Suppose that f:GγC is a continuous function that is analytic on G, f(G)Ω and f(γ)ω, then for any compact set κγ, f has an analytic continuation to an open set containing Gκ.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
  • 2 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Schwarz reflection principle
Canonical name SchwarzReflectionPrinciple
Date of creation 2013-03-22 14:17:58
Last modified on 2013-03-22 14:17:58
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 30C35
Synonym Schwarz reflection theorem
Synonym reflection principle
Defines symmetric region