Schwarz Reflection Principle

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/SchwarzReflectionPrinciple.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SchwarzReflectionPrinciple.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Todd Rowland

Example

Schwarz reflection for z+((z^2)/2)

First some graphics are premade (problems with DisplayIdentity in FilledListPlot).

<<Graphics`Colors`;
<<Graphics`FilledPlot`;
<<Graphics`ComplexMap`;

tt=PolarMap[(#1+#1^2/2&),{0,1},{0,Pi},DisplayFunction->Identity]
tt2=PolarMap[(#1+#1^2/2&),{0,1},{Pi,2Pi},DisplayFunction->Identity]

Clear[ss,ss2];
ss=FilledListPlot[Table[{Cos[t]+Cos[2t]/2,Sin[t]+Sin[2t]/2},{t,Pi,0,-Pi/40}],Fills->Red]
ss2=FilledListPlot[Table[{Cos[t]+Cos[2t]/2,-Sin[t]-Sin[2t]/2},{t,Pi,0,-Pi/40}],Fills->Pink]

Clear[rr,rr2];
rr=PolarMap[Identity,{0,1},{0,Pi}]
rr2=PolarMap[Identity,{0,1},{Pi,2Pi}]

Clear[uu];
uu=StringJoin["w = f(",ToString[ z,TraditionalForm],")"]

The first diagram shows the original domain, the upper half disk.

Show[{Graphics[{Red,Disk[{0,0},1,{0,Pi}]}]
,rr},AspectRatio->Automatic,Axes->True,
AxesLabel->{TraditionalForm[z],None},
Ticks->False,PlotRange->{{-1,1.2},{-1.1,1.1}}]

The second diagram shows the image of the original domain by the function f.  In general, all that is necessary is for the real axis to be mapped to the real axis.  The rest of the function does not have to stay on one side, e.e. z+z^2 or cos z.  The function z+ z^2/2 was chosen so that it did stay on one side, and made a clearer picture.

Show[{ss,Graphics[Text[uu,{1.7,-.2}]],tt},Ticks->False,
PlotRange->{{-1,2},{-1.3,1.3}},AspectRatio->Automatic]

The third diagram shows the original domain and its reflection in pink.

gr1=Show[{Graphics[{Red,Disk[{0,0},1,{0,Pi}],Pink,Disk[{0,0},1,{Pi,2Pi}]

}],rr,rr2},AspectRatio->Automatic,Axes->True,
AxesLabel->{TraditionalForm[z],None},
Ticks->False,PlotRange->{{-1,1.2},{-1.1,1.1}}]

The fourth diagram shows the image of the relcted domain reflected, i.e., the image of the reflected domain values.

gr2=Show[{ss,ss2,Graphics[Text[TraditionalForm[uu],{1.7,-.7}]],tt,tt2},Ticks->False,
PlotRange->{{-1,2},{-1.3,1.3}},AspectRatio->Automatic]

Show[GraphicsArray[{gr1,gr2}]]