Analytic Continuation

Author

Eric W. Weisstein
April 15, 2004

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

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

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

Additional Authors

Code and figured by
Todd Rowland
February 25, 2002

Analytic Continuation of (z^(1/2)).

The first diagram shows the disk of the radius of convergence for the analytic continuation of the square root of z along the path of the unit circle.  The power series representation is only valid in the disk.  The power series for other points on the path is given either by differentiating, or by substitution.  This is a fortunate situation, in that the radius of convergence remains 1 and the function can be continued along the path.

<<Graphics`Colors`
w0[t_]:=E^(I t/2)
w1[t_]:=1/2 E^(-I t/2)
w2[t_]:=-1/8 E^(-3I t/2)

movie=Table[Graphics[{PointSize[.03],
Pink,Disk[{Cos[t],Sin[t]},1],
Red,Circle[{Cos[t],Sin[t]},1],
Blue,Point[{Cos[t],Sin[t]}],{Black,Circle[{0,0},1]},
{Black,Point[{0,0}]}},PlotRange->{{-2,2},{-2,2}},AspectRatio->Automatic],
{t,0,2Pi,Pi/20}];

Export[gifs<>"analcont.gif",Show/@Drop[movie,-1]]

Length[movie]

Show[GraphicsArray[partition[Take[movie,{1,41,3}],7]]]

The second animation shows the value of the function being continued along with a couple of its derivatives, which correspond to terms in the power series expansion.  Of course, analytically continuing the function around the circle brings the function to its negative, the other square root.  I had trouble combining the two animations into one, since ideally they should be synchronized.

Table[Show[Graphics[Text[StyleForm[TableForm[{{"(Subscript[z, 0]) = " ,PaddedForm[N[ (E^(I t))],{2,2}]},{"f  =",PaddedForm[N[w0[t]],{2,2}]},{"f' = ",PaddedForm[N[w1[t]],{2,2}]},{"f'' =",PaddedForm[N[2w2[t]],{2,2}]}}],FontSize->20]
,{0,0},FormatType->TraditionalForm]
],
PlotRange->{{-1,1},{-1,1}},AspectRatio->Automatic],
{t,0,2Pi,Pi/20}]

Another animation for Analytic Continuation

Analytic continuation of the square root function around the unit circle

In this animation, the domain space is filled in with pink on the left.  It gets mapped to the blue region on the right by the square root function.  The light blue region is the negative, or other square root.  However, by continuing the function around the circle, the square root function takes values in what used to be the light blue region.  The roles of the blue and light blue region are reversed.  This can be interpreted as going from one branch to the other, for the multivalued function square root.

The point is that, for analytic continuation, the function is extended using the nearby values.  These provide the information on the power series (see old animation) to write the analytic function centered around the point, which moves around the circle.

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

Clear[costh,qu];
costh[t_]:= 2^(-1/2)(1+Cos[t])^(1/2)
qu[t_]:=(2+2Cos[t])^(1/4){((costh[t]+1)/2)^(1/2),Sign[Pi-t]((1-costh[t])/2)^(1/2)}
qu[t_,th_]:={{Cos[th],-Sin[th]},{Sin[th],Cos[th]}}.qu[t]
Clear[f2,w0];
w0[t_]:=E^(I t/2)
f2[t_][z_]:= w0[t]Sqrt[1+z]


pf[t_]:=PolarMap[f2[t],{0,1},{0,2Pi},PlotRange->1.5{{-1,1},{-1,1}},Ticks->False,DisplayFunction->Identity]


sir[t_]:=Graphics[{{Black,Point[{0,0}]},
{PointSize[.03],
Pink,Disk[{Cos[t],Sin[t]},1],
Red,Circle[{Cos[t],Sin[t]},1],
Blue,Point[{Cos[t],Sin[t]}]},{Black,Circle[{0,0},1]}},
AspectRatio ->Automatic,Axes->True,Ticks->False,PlotRange->2{{-1,1},{-1,1}}
]

redub[th_,clr_,n_Integer]:=
Graphics[{clr,Polygon[Table[qu[t,th/2],{t,0,2Pi,Pi/n}]]},AspectRatio ->Automatic,Axes->True,Ticks->False,PlotRange->1.5{{-1,1},{-1,1}}]

dub[t_]:=
Graphics[{Rectangle[{0,0},{1,1},redub[t,Blue,20]],Rectangle[{0,0},{1,1},pf[t]],
Rectangle[{0,0},{1,1},redub[t+2Pi,LightBlue,20]],Rectangle[{0,0},{1,1},pf[t+2Pi]]
},AspectRatio->Automatic,PlotRange->{{0,1},{0,1}}]

movie=Table[GraphicsArray[{sir[s],dub[s]}],{s,0,4Pi-Pi/20,Pi/20}];

Export[gifs<>"analcont.gif",movie]//Timing

Length[movie]

Show[GraphicsArray[Partition[Take[movie,{1,80,10}],4]]]