Schwarz function

The Schwarz function of a curve in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> is an <a href="/facts/Complex_analytic_function/kVzthtEf">analytic function</a> which maps the points of the curve to their <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugates</a>. It can be used to generalize the <a href="/facts/Schwarz_reflection_principle/gsxRVe13">Schwarz reflection principle</a> to reflection across arbitrary <a href="/facts/Analytic_curve/JhL3z8FP">analytic curves</a>, not just across the real axis.
The Schwarz function exists for analytic curves. More precisely, for every non-singular, analytic <a href="/facts/Jordan_arc/zKpty4QT">Jordan arc</a> 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 in the complex plane, there is an <a href="/facts/Open_neighborhood/XHKgrpkp">open neighborhood</a> 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 of 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 and a unique analytic function 
 
 
 
 S
 
 
 {\displaystyle S}
 
 on 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 such that 
 
 
 
 S
 (
 z
 )
 =
 
 
 z
 ¯
 
 
 
 
 {\displaystyle S(z)={\overline {z}}}
 
 for every 
 
 
 
 z
 ∈
 Γ
 
 
 {\displaystyle z\in \Gamma }
 
.
The "Schwarz function" was named by <a href="/facts/Philip_J._Davis/VMXtMMpJ">Philip J. Davis</a> and <a href="/facts/Henry_O._Pollak/C7wuNHIA">Henry O. Pollak</a> (1958) in honor of <a href="/facts/Hermann_Schwarz/lP1VRdjH">Hermann Schwarz</a>, who introduced the Schwarz reflection principle for analytic curves in 1870. However, the Schwarz function does not explicitly appear in Schwarz's works.

Schwarz function open-in-new

Schwarz function