Inside-Outside Theorem

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/Inside-OutsideTheorem.nb.

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

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

Additional Authors

Code and description by
Todd Rowland

Explanation

The diagram shows two different points of view of the contour γ andthe poles of (P/Q) dzon the Riemann sphere.  The usual point of view is centered at z=0, but the role of inside and outside is switched from the point of view of z=∞.  The poles inside are labeled blue and outside are green.

<<Graphics`Colors`;
<<Graphics`Arrow`;

Clear[adarrs];
adarrs[a_List,n_Integer]:=Module[{pieces,l=Length[a],k},
k=Floor[l/n];
pieces=Partition[a,k];
(Arrow[#1[[1]],#1[[2]]]&)/@pieces]

Clear[aa,qu,oneover,bb,cc];
oneover[{x_,y_}]:={x,-y}/(x^2+y^2)

aa=Table[(5+Sin[12Pi i/60+Pi/5]+2Sin[Pi i/60])
{Cos[2Pi i/60],Sin[2Pi i/60]},{i,0,60}];
bb={{2,2},{-3,-2}};
cc={{-6,7},{5,6}};
qu[m_,a_List]:=Module[{l=Length[a]},
{{AbsolutePointSize[5],Point[{0,0}]},{Red,Line[a]},{Red,adarrs[a,m]}}]

Show[GraphicsArray[{Graphics[{Text[TraditionalForm[HoldForm[z=0]],{0,0},{0,2}],
{Blue,AbsolutePointSize[5],Point/@bb,Green,Point/@cc},
qu[3,aa]},AspectRatio-> Automatic,PlotRange->{{-8,8},{-8,8}}],
Graphics[{Text[TraditionalForm[HoldForm[z=∞]],{0,0},{0,2}],
{Blue,AbsolutePointSize[5],Point/@(oneover/@bb),Green,Point/@(oneover/@cc)},
qu[3,oneover/@aa]},AspectRatio-> Automatic,PlotRange->{{-.3,.3},{-.3,.3}}]}]]