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Mathematica "prefers" complex numbers to real numbers in various ways--except unfortunately when it comes to plotting, where it expects you to break things apart into real and complex parts. But if you apply David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html), then you may work directly with complex numbers in plotting. And, as in this example, let Mathematica do the work of showing that the image points lie on a circle:

<<Presentations`
F[z_] := (5 -I z)/(5^2 + z^2)
pts = {-7, -2, 0, 2, 7};
xAxis = ComplexCurve[-7(1-t) + 7t, {t, 0, 1}];

Draw2D[{
        {Blue, xAxis, PointSize[Large], Red, ComplexPoint/@pts} // ComplexMap[F]
       },
       Axes -> True,
       PlotRange->ComplexPlotRange[-0.05-0.15I,0.25+0.15I],
       ImageSize->Scaled[0.45]]

Mathematica "prefers" complex numbers to real numbers in various ways  -- except unfortunately when it comes to plotting, where it expects you to break things apart into real and complex parts. But if you apply David Park's Presentations add-on, then you may work directly with complex numbers in plotting. And, as in this example, let   Mathematica do the work of showing that the image points lie on a circle:

<<Presentations`
F[z_] := (5 - I z)/(5^2 + z^2)
pts = {-7, -2, 0, 2, 7};
xAxis = ComplexCurve[ -7 ( 1 - t ) + 7 t, {t, 0, 1}];

Draw2D[{
        { Blue, xAxis, PointSize[Large], Red, ComplexPoint /@ pts} // ComplexMap[F]
       },
       Axes -> True,
       PlotRange -> ComplexPlotRange[-0.05 - 0.15 I, 0.25 + 0.15 I ],
       ImageSize -> Scaled[0.45] ]

Mathematica "prefers" complex numbers to real numbers in various ways--except unfortunately when it comes to plotting, where it expects you to break things apart into real and complex parts. But if you apply David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html), then you may work directly with complex numbers in plotting. And, as in this example, let Mathematica do the work of showing that the image points lie on a circle:

<<Presentations`
F[z_] := (5 -I z)/(5^2 + z^2)
pts = {-7, -2, 0, 2, 7};
xAxis = ComplexCurve[-7(1-t) + 7t, {t, 0, 1}];

Draw2D[{
        {Blue, xAxis, PointSize[Large], Red, ComplexPoint/@pts} // ComplexMap[F]
       },
       Axes -> True]

Mathematica "prefers" complex numbers to real numbers in various ways--except unfortunately when it comes to plotting, where it expects you to break things apart into real and complex parts. But if you apply David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html), then you may work directly with complex numbers in plotting. And, as in this example, let Mathematica do the work of showing that the image points lie on a circle:

<<Presentations`
F[z_] := (5 -I z)/(5^2 + z^2)
pts = {-7, -2, 0, 2, 7};
xAxis = ComplexCurve[-7(1-t) + 7t, {t, 0, 1}];

Draw2D[{
        {Blue, xAxis, PointSize[Large], Red, ComplexPoint/@pts} // ComplexMap[F]
       },
       Axes -> True,
       PlotRange->ComplexPlotRange[-0.05-0.15I,0.25+0.15I],
       ImageSize->Scaled[0.45]]

Mathematica "prefers" complex numbers to real numbers in various ways--except unfortunately when it comes to plotting, where it expects you to break things apart into real and complex parts. But if you apply David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html), then you may work directly with complex numbers in plotting. And, as in this example, let Mathematica do the work of showing that the image points lie on a circle:

<<Presentations`
F[z_] := (5 -I z)/(5^2 + z^2)
pts = {-7, -2, 0, 2, 7};
xAxis = ComplexCurve[-7(1-t) + 7t, {t, 0, 1}];

Draw2D[{
        {Blue, xAxis, PointSize[Large], Red, ComplexPoint/@pts} // ComplexMap[F]
       },
       Axes -> True]

The output (which I'm unable to upload for technical reasons having nothing to do with Mathematica) is essentially the same as what you see in the first plot from @chris's answer.

Mathematica "prefers" complex numbers to real numbers in various ways--except unfortunately when it comes to plotting, where it expects you to break things apart into real and complex parts. But if you apply David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html), then you may work directly with complex numbers in plotting. And, as in this example, let Mathematica do the work of showing that the image points lie on a circle:

<<Presentations`
F[z_] := (5 -I z)/(5^2 + z^2)
pts = {-7, -2, 0, 2, 7};
xAxis = ComplexCurve[-7(1-t) + 7t, {t, 0, 1}];

Draw2D[{
        {Blue, xAxis, PointSize[Large], Red, ComplexPoint/@pts} // ComplexMap[F]
       },
       Axes -> True]