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Notebook[{

Cell[CellGroupData[{
Cell["Winding Numbers", "Subsection"],

Cell[CellGroupData[{

Cell["Example", "Subsubsection"],

Cell[TextData[{
  "Calculate the winding number of the image of circles of radii 1/2, 1 and 2 \
under the map ",
  Cell[BoxData[
      \(TraditionalForm\`f(z)\  = \ \((z\^3 + 2)\)/z\)]],
  "."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solution ", "Subsubsection"],

Cell[TextData[{
  "The winding number equals the number of zeros ",
  Cell[BoxData[
      \(TraditionalForm\`-\)]],
  " the number of poles inside the curve. The zeros of ",
  Cell[BoxData[
      \(TraditionalForm\`f(z)\)]],
  " are the cube roots of ",
  Cell[BoxData[
      \(TraditionalForm\`\(-2\)\)]],
  ". Since,"
}], "Text"],

Cell[BoxData[
    \(N[\@2\%3]\)], "Input"],

Cell[TextData[{
  "all three zeros lie inside the circle of radius 2, but none lie inside the \
circles of radii 1/2 or 1.  ",
  Cell[BoxData[
      \(TraditionalForm\`f(z)\)]],
  " has a simple pole at the origin which lies in all three circles.\n\
Therefore, the winding numbers for the images of the circles of radii 1/2 and \
1 are both ",
  Cell[BoxData[
      \(TraditionalForm\`0 - 1 = \(-1\)\)]],
  ". The winding number for the image of the circle of radius 2 is ",
  Cell[BoxData[
      \(TraditionalForm\`3\  - \ 1\  = \ 2\)]],
  ".\nHere are some graphical representations."
}], "Text"],

Cell[BoxData[{
    \(\(z\  = \ x\  + \ I\ y;\)\), "\n", 
    \(u = Re[\((z\^3\  + \ 2)\)/z] // ComplexExpand; \), "\n", 
    \(\(v = Im[\((z\^3\  + \ 2)\)/z] // ComplexExpand;\)\), "\n", 
    \(x\  = \ r\ Cos[t]; \ y\  = \ r\ Sin[t];\)}], "Input"],

Cell[TextData[{
  StyleBox["The image of thecircle of radius 1/2",
    FontSlant->"Italic"],
  " (animate the graphic by selecting it and pressingCTRL-Y)."
}], "Text"],

Cell[BoxData[{
    \(\(r\  =  .5;\)\), "\[IndentingNewLine]", 
    \(Do[ParametricPlot[{u, v}, {t, 0, p}, 
        PlotRange \[Rule] {{\(-5\), 5}, {\(-5\), 5}}], {p, \[Pi]/4, 
        2  \[Pi], \[Pi]/4}]\)}], "Input"],

Cell[TextData[{
  "The curve circles the origin counterclockwise once, so the winding number \
is ",
  Cell[BoxData[
      \(TraditionalForm\`\(-1\)\)]],
  ".\n\n",
  StyleBox["The image of the circle of radius 1.",
    FontSlant->"Italic"]
}], "Text"],

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    \(\(r\  = 1;\)\), "\[IndentingNewLine]", 
    \(Do[ParametricPlot[{u, v}, {t, 0, p}, 
        PlotRange \[Rule] {{\(-5\), 5}, {\(-5\), 5}}], {p, \[Pi]/4, 
        2  \[Pi], \[Pi]/4}]\)}], "Input"],

Cell[TextData[{
  "The curve still circles the origin counterclockwise once, so the winding \
number is ",
  Cell[BoxData[
      \(TraditionalForm\`\(-1\)\)]],
  ".\n\n",
  StyleBox["The image of the circle of radius 2.",
    FontSlant->"Italic"]
}], "Text"],

Cell[BoxData[{
    \(\(r\  = 2;\)\), "\[IndentingNewLine]", 
    \(Do[ParametricPlot[{u, v}, {t, 0, p}, 
        PlotRange \[Rule] {{\(-5\), 5}, {\(-5\), 5}}], {p, \[Pi]/4, 
        2  \[Pi], \[Pi]/4}]\)}], "Input"],

Cell[TextData[{
  "Now the curve circles the origin twice in a counter-clockwise direction, \
so the winding number is 2.\nNext are images of the circles of ",
  Cell[BoxData[
      \(TraditionalForm\`r\  = \ \ 1/2\)]],
  " to 2 in steps of .1. You can see parts of the curve cross over the origin \
when ",
  Cell[BoxData[
      \(TraditionalForm\`r\)]],
  " passes through the cube root of 2, changing the winding number from ",
  Cell[BoxData[
      \(TraditionalForm\`\(-1\)\)]],
  " to 2."
}], "Text"],

Cell[BoxData[
    \(Do[ParametricPlot[{u, v}, {t, 0, 2  \[Pi]}, 
        PlotRange \[Rule] {{\(-5\), 5}, {\(-5\), 5}}], {r,  .5, 
        2,  .1}]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example", "Subsubsection"],

Cell[TextData[{
  "Calculate the winding number of the image of circles of radii 1/4 and 1 \
under the map ",
  Cell[BoxData[
      \(TraditionalForm\`f(z)\  = \ \((2  z - 1)\)\^7/z\^3\)]],
  "."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solution ", "Subsubsection"],

Cell[TextData[{
  "The winding number equals the number of zeros ",
  Cell[BoxData[
      \(TraditionalForm\`-\)]],
  " the number of poles inside the curve. ",
  Cell[BoxData[
      \(TraditionalForm\`f(z)\)]],
  " has a zero of order 7 at  ",
  Cell[BoxData[
      \(TraditionalForm\`1/2\)]],
  ", and a pole of order three at the origin. Therefore, the winding number \
of the image of the circle of radius 1/4 is ",
  Cell[BoxData[
      \(TraditionalForm\`0 - 3 = \(-3\)\)]],
  ", and the winding number of the image of the circle of radius 1 is ",
  Cell[BoxData[
      \(TraditionalForm\`7 - 3 = 4\)]],
  "."
}], "Text"],

Cell["Here are some graphical representations.", "Text"],

Cell[BoxData[{
    \(\(z\  = \ x\  + \ I\ y;\)\), "\n", 
    \(u = Re[\((2  z - 1)\)\^7/z\^3] // ComplexExpand; \), "\n", 
    \(\(v = Im[\((2  z - 1)\)\^7/z\^3] // ComplexExpand;\)\), "\n", 
    \(x\  = \ r\ Cos[t]; \ y\  = \ r\ Sin[t];\)}], "Input"],

Cell[TextData[{
  StyleBox["The image of the circle of radius 1/4.",
    FontSlant->"Italic"],
  "\nThe start of the curve shows that it is moving clockwise."
}], "Text"],

Cell[BoxData[{
    \(\(r\  =  .25;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, \[Pi]/2}, 
      PlotRange \[Rule] {{\(-15\), 10}, {\(-5\), 5}}]\)}], "Input"],

Cell["The curve wraps around twice near the origin:", "Text"],

Cell[BoxData[{
    \(\(r\  =  .25;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, 2  \[Pi]}, 
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Cell["and once more much further away:", "Text"],

Cell[BoxData[{
    \(\(r\  = 1;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, 2  \[Pi]}, 
      PlotRange \[Rule] {{\(-500\), 2500}, {\(-1500\), 1500}}]\)}], "Input"],

Cell[TextData[{
  "Thus, the winding number is ",
  Cell[BoxData[
      \(TraditionalForm\`\(-3\)\)]],
  ".\n\n",
  StyleBox["The image of the circle of radius 1.",
    FontSlant->"Italic"],
  "\nThe start of the curve shows that it is moving counter-clockwise."
}], "Text"],

Cell[BoxData[{
    \(\(r\  = 1;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, \[Pi]/2}, 
      PlotRange \[Rule] {{\(-10\), 15}, {\(-5\), 5}}]\)}], "Input"],

Cell["The curve wraps around twice near the origin:", "Text"],

Cell[BoxData[{
    \(\(r\  = 1;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, 2  \[Pi]}, 
      PlotRange \[Rule] {{\(-10\), 15}, {\(-5\), 5}}]\)}], "Input"],

Cell["once more further away:", "Text"],

Cell[BoxData[{
    \(\(r\  = 1;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, 2  \[Pi]}, 
      PlotRange \[Rule] {{\(-300\), 100}, {\(-100\), 100}}]\)}], "Input"],

Cell["and a final loop very far from the origin:", "Text"],

Cell[BoxData[{
    \(\(r\  = 1;\)\), "\[IndentingNewLine]", 
    \(ParametricPlot[{u, v}, {t, 0, 2  \[Pi]}, 
      PlotRange \[Rule] {{\(-500\), 2500}, {\(-1500\), 1500}}]\)}], "Input"],

Cell["Thus, the total winding number is 4.", "Text"]
}, Open  ]]
}, Open  ]]
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