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0\)]], " is to find a function ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], " that is harmonic in the upper half plane and has the boundary values ", Cell[BoxData[ \(\[Phi] \((x, 0)\) = U \((x)\)\)]], ", where ", Cell[BoxData[ \(U \((x)\)\)]], ", is a real-valued function of the real variable x. An important method \ for solving this problem is our next result which is attributed to the French \ mathematician ", ButtonBox["Sim\[EAcute]on Poisson", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poisson.\ html"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ StyleBox["Theorem 10.3 (", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ButtonBox["Poisson's Integral Formula", ButtonData:>{ URL[ "http://mathworld.wolfram.com/PoissonsHarmonicFunctionFormula.html"], None}, ButtonStyle->"Hyperlink"], StyleBox["), Page 401.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " Let U be a real-valued function that is piecewise continuous and bounded \ for all real t. The function \n\n\t\t\t", Cell[BoxData[ RowBox[{\(\[Phi] \((x, y)\)\), "=", RowBox[{\(y\/\[Pi]\), " ", RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(\(U \((t)\)\)\/\(\((x\ - \ t)\)\^2\ + \ y\^2\)\) \[DifferentialD]t\)}]}]}]]], " \n\nis harmonic in the upper half plane ", Cell[BoxData[ \(Im[z] > 0\)]], " and has the boundary values \n\n\t\t\t", Cell[BoxData[ \(\[Phi] \((x, 0)\) = U \((x)\)\)]], " \n\nwherever U is continuous." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Proof of Theorem 10.3, see text Page 401.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], StyleBox["Example 10.11, Page 403.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find a function ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], " that is harmonic in the upper half-plane ", Cell[BoxData[ \(Im[z] > 0\)]], ", which takes on the boundary values \n\n", Cell[BoxData[ \(\[Phi] \((x, 0)\) = 1\ \ \ when\ \ \ | x | \(\(<\)\(1\)\), \[IndentingNewLine]\[Phi] \((x, 0)\) = 0\ \ \ when\ \ \ | x | \(\(>\)\(1.\)\)\)]], " " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 10.11.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "The solution is similar to Example 10.7, Page 316, but the method of \ solution is different.\n\nEnter the function U[t] and use the Poisson \ integral to construct ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[U, v, x, y, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[t, \[Phi]];\)\ \), "\n", \(\(U[t_]\ = \ 1;\)\ \), "\n", \(\(\[Phi][x_, y_]\ = \ Apart[Together[ y\/\[Pi]\ \(\[Integral]\_\(-1\)\%1\( U[t]\/\(\((x - t)\)\^2 + y\^2\)\) \[DifferentialD]t\)]];\)\ \), "\n", \(\(v[x_, y_]\ = \ ReplaceAll[\[Phi][x, y], ArcTan[z_] \[Rule] \(-ArcTan[1\/z]\)];\)\ \), "\n", \(\(Print["\< U[t] = \>", U[t]];\)\ \ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print[\*"\"\<\[Phi][x,y] = \!\(y\/\[Pi]\) \ \!\(\[Integral]\_\(-1\)\%1\)\!\(U[t]\/\(\((x - t)\)\^2 + y\^2\)\)\ \[DifferentialD]t\>\""];\)\ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", v[x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nUsing the trigonometric identity ", Cell[BoxData[ \(\(\(\ \)\(arctan \((\(-t\))\) = \(-arctan\) \((t)\)\)\)\)]], ", the above result can be written as ", Cell[BoxData[ \(\[Phi] \((x, y)\) = \(1\/\[Pi]\) arctan \((y\/\(x - 1\))\) - \(1\/\[Pi]\) arctan \((y\/\(x + 1\))\)\)]], ". We can verify some of the boundary values by taking limits." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print["\< \[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\n", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][-2,y] = \>\"", Limit[\[Phi][\(-2\), y], \ y \[Rule] 0]];\)\ \), "\n", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(\(-1\)\/2\),y] = \ \>\"", Limit[\[Phi][\(-1\)\/2, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][0,y] = \>\"", Limit[\[Phi][0, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(1\/2\),y] = \ \>\"", Limit[\[Phi][1\/2, y], \ y \[Rule] 0]];\)\ \), "\n", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][2,y] = \>\"", Limit[\[Phi][2, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nUse ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a contour plot of the solution." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(cont = Table[k, {k, 0, 1, 1\/10}];\)\ \), "\n", \(\(ContourPlot[\[Phi][x, y], {x, \(-2\), 2}, {y, 0.000001, 2}, \[IndentingNewLine]Contours \[Rule] cont, PlotPoints \[Rule] 35, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]ColorFunction \[Rule] Hue];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print[\[Phi][x, y] \[Equal] c];\)\ \), "\n", \(\(Print["\", cont];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nThen use ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a 3D plot of the solution." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Plot3D[\[Phi][x, y], {x, \(-2\), 2}, {y, 0.00001, 1.5}, PlotPoints \[Rule] {41, 16}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\", "\<\ \[Phi][x,y] \>"}, ColorFunction \[Rule] Hue];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nTherefore, the function ", Cell[BoxData[ \(\[Phi][x, y] = \(1\/\[Pi]\) ArcTan[\(1 - x\)\/y] + \(1\/\[Pi]\) ArcTan[\(1 + x\)\/y]\)]], " is harmonic in the upper half-plane ", Cell[BoxData[ \(Im[z] > 0\)]], ", and takes on the desired boundary values." }], "Text"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Example 10.12, Page 403.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find a function ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], " that is harmonic in the upper half-plane ", Cell[BoxData[ \(Im[z] > 0\)]], ", which takes on the boundary values \n\n", Cell[BoxData[ \(\[Phi] \((x, 0)\) = x\ \ \ when\ \ \ | x | \(\(<\)\(1\)\), \[IndentingNewLine]\[Phi] \((x, 0)\) = 0\ \ \ when\ \ \ | x | \(\(>\)\(1.\)\)\)]], " \n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 10.12.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "Enter the function U[t] and use the Poisson integral to construct ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[t, U, v, x, y, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[\[Phi]];\)\ \), "\n", \(\(U[t_]\ = \ t;\)\ \), "\[IndentingNewLine]", \(\(g[t_] = y\/\[Pi]\ \(\[Integral]\(U[t]\/\(\((x - t)\)\^2 + y\^2\)\) \[DifferentialD]t\);\)\ \), "\n", \(\(\[Phi][x_, y_]\ = \ Expand[y\/\[Pi]\ \(\[Integral]\_\(-1\)\%1\( U[t]\/\(\((x - t)\)\^2 + y\^2\)\) \[DifferentialD]t\)];\)\ \), "\ \[IndentingNewLine]", \(\(\[Phi][x_, y_]\ = \ Expand[g[1] - g[\(-1\)]];\)\ \), "\n", \(\(v[x_, y_]\ = ReplaceAll[\[Phi][x, y], ArcTan[z_] \[Rule] \(-ArcTan[1\/z]\)];\)\ \), "\n", \(\(Print["\< U[t] = \>", U[t]];\)\ \ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print[\*"\"\<\[Phi][x,y] = \!\(y\/\[Pi]\) \ \!\(\[Integral]\_\(-1\)\%1\)\!\(U[t]\/\(\((x - t)\)\^2 + y\^2\)\)\ \[DifferentialD]t\>\""];\)\ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", v[x, y]];\)\ \), "\n", \(\)}], "Input"], Cell[TextData[{ "\nUsing the identities ", Cell[BoxData[ \(arctan \((\(-t\))\) = \(\(-arctan\) \((t)\)\ \ and\ \ ln \((a\/b)\) = ln \((a)\) - ln \((b)\)\)\)]], ", the above result can be written as ", Cell[BoxData[ \(\[Phi] \((x, y)\) = \(x\/\[Pi]\) arctan \((y\/\(x\ - \ 1\))\) - \(x\/\[Pi]\) arctan \((y\/\(x\ + \ 1\))\) + \(y\/\(2\ \[Pi]\)\) ln \((\(\((x - 1)\)\^2\ + \ y\^2\)\/\(\((x + 1)\)\^2\ + \ \ y\^2\))\)\)]], ". However for computing values of ArcTan we use the two variable form of \ the function and the following version of ", Cell[BoxData[ \(\[Phi][x, y]\)]], ". We can verify some of the boundary values by taking limits." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(\[Phi][x_, y_] = \(x\/\[Pi]\) ArcTan[x - 1, y] - \(x\/\[Pi]\) ArcTan[x + 1, y] + \(y\/\(2\ \[Pi]\)\) Log[\(\((x - 1)\)\^2 + y\^2\)\/\(\((x + 1)\)\^2 + y\^2\)];\)\ \), \ "\[IndentingNewLine]", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\n", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(\(-3\)\/2\),y] = \ \>\"", Limit[\[Phi][\(-3\)\/2, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(\(-3\)\/4\),y] = \ \>\"", Limit[\[Phi][\(-3\)\/4, y], \ y \[Rule] 0]];\)\ \), "\n", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(\(-1\)\/2\),y] = \ \>\"", Limit[\[Phi][\(-1\)\/2, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][0,y] = \>\"", Limit[\[Phi][0, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(1\/2\),y] = \ \>\"", Limit[\[Phi][1\/2, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(3\/4\),y] = \ \>\"", Limit[\[Phi][3\/4, y], \ y \[Rule] 0]];\)\ \), "\n", \(\(Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \[Phi][\!\(3\/2\),y] = \ \>\"", Limit[\[Phi][3\/2, y], \ y \[Rule] 0]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nUse ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a contour plot of the solution." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(ContourPlot[\[Phi][x, y], {x, \(-2\), 2}, {y, 0.000001, 2}, \[IndentingNewLine]Contours \[Rule] 10, PlotPoints \[Rule] 35, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]ColorFunction \[Rule] Hue];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nThen use ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a 3D plot of the solution." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Plot3D[\[Phi][x, y], {x, \(-1.5\), 1.5}, {y, 10\^\(-8\), 0.5}, PlotPoints \[Rule] {31, 11}, \[IndentingNewLine]PlotRange \[Rule] {{\(-1.5\), 1.5}, {0, 0.5}, {\(-1.05\), 1.05}}, AxesLabel \[Rule] {"\", "\< y\>", "\<\[Phi][x,y] \>"}, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[0, 0.5, 0.25], Range[\(-1\), 1, 0.5]}, BoxRatios \[Rule] {1, 1, 0.8}, ViewPoint \[Rule] {2.5, \(-3\), 2}, ColorFunction \[Rule] Hue];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nTherefore, the function ", Cell[BoxData[ \(\(\(\ \)\(\[Phi] \((x, y)\) = \(x\/\[Pi]\) arctan \((y\/\(x\ - \ 1\))\) - \(x\/\[Pi]\) arctan \((y\/\(x\ + \ 1\))\) + \(y\/\(2\ \[Pi]\)\) ln \((\(\((x - 1)\)\^2\ + \ y\^2\)\/\(\((x + 1)\)\^2\ + \ \ y\^2\))\)\)\)\)]], " is harmonic in the upper half-plane ", Cell[BoxData[ \(Im[z] > 0\)]], ", and takes on the desired boundary values." }], "Text"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Example 10.13, Page 404.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Use Poisson's Integral formula to find the harmonic function ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], " that is harmonic in the upper half-plane ", Cell[BoxData[ \(Im[z] > 0\)]], ", that takes on the boundary values \n\n", Cell[BoxData[ \(\[Phi] \((x, 0)\) = \(-1\)\ \ \ for\ \ \ \ \ \ \ x < \(-1\), \ \[IndentingNewLine]\[Phi] \((x, 0)\) = \ x\ \ \ \ for\ \ \ - 1 < x < 1, \[IndentingNewLine]\[Phi] \((x, 0)\) = \ 1\ \ \ \ for\ \ \ \ \ \ \ 1 < \(\(x\)\(.\)\)\)]], " " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 10.13.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "This is similar to Example 10.17, Page 416, but the method of solution is \ different.\n\nUsing techniques from ", ButtonBox["Section 10.2", ButtonData:>{"ca1002.nb", None}, ButtonStyle->"Hyperlink"], ", the function ", Cell[BoxData[ \(v \((x, y)\) = 1 - \(1\/\[Pi]\) arctan \((y\/\(x\ + \ 1\))\) - \(1\/\[Pi]\) arctan \((y\/\(x\ - \ 1\))\)\)]], " is harmonic in the upper half plane and takes on the boundary values\n", Cell[BoxData[ \(v \((x, 0)\) = \(-1\)\ \ \ for\ \ \ \ \ \ \ x < \(-1\), \ \[IndentingNewLine]v \((x, 0)\) = \ 0\ \ \ \ for\ \ \ - 1 < x < 1, \[IndentingNewLine]v \((x, 0)\) = \ 1\ \ \ \ for\ \ \ \ \ \ \ 1 < \(\(x\)\(.\)\)\)]], " \n\nThus, we should add it to the solution ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], " in Example 10.12 to obtain the desired result. However, with ", StyleBox["Mathematica", FontSlant->"Italic"], " we need to use ", Cell[BoxData[ \(ArcTan[x \[PlusMinus] 1, y]\ \ instead\ of\ \ ArcTan[ y\/\(x\ \[PlusMinus] \ 1\)]\)]], ".\n\nEnter the function U[t] and use the Poisson integral to construct ", Cell[BoxData[ \(\[Phi] \((x, y)\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[t, U, v, V, x, y, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[\[Phi]];\)\ \), "\n", \(\(U[t_]\ = \ t;\)\ \), "\[IndentingNewLine]", \(\(g[t_] = y\/\[Pi]\ \(\[Integral]\(U[t]\/\(\((x - t)\)\^2 + y\^2\)\) \[DifferentialD]t\);\)\ \), "\n", \(\(\[Phi][x_, y_]\ = \ Expand[y\/\[Pi]\ \(\[Integral]\_\(-1\)\%1\( U[t]\/\(\((x - t)\)\^2 + y\^2\)\) \[DifferentialD]t\)];\)\ \), "\ \[IndentingNewLine]", \(\(\[Phi][x_, y_]\ = \ Expand[g[1] - g[\(-1\)]];\)\ \), "\n", \(\(v[x_, y_]\ = \ ReplaceAll[\[Phi][x, y], ArcTan[z_] \[Rule] \(-ArcTan[1\/z]\)];\)\ \), "\n", \(\(V[x_, y_]\ = \ 1\ - \ \(1\/\[Pi]\) ArcTan[x + 1, y]\ - \(1\/\[Pi]\) ArcTan[x - 1, y];\)\ \), "\n", \(\(\[Phi][x_, y_]\ = \ \[Phi][x, y]\ + \ V[x, y];\)\ \), "\n", \(\(v[x_, y_]\ = \ v[x, y]\ + \ V[x, y];\)\ \), "\n", \(\(Print["\< U[t] = \>", U[t]];\)\ \ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\", V[x, y]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print[\*"\"\<\[Phi][x,y] = \!\(y\/\[Pi]\) \ \!\(\[Integral]\_\(-1\)\%1\)\!\(U[t]\/\(\((x - t)\)\^2 + y\^2\)\)\ \[DifferentialD]t + V[x,y]\>\""];\)\ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", v[x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nUse ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a contour plot of the solution." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\[Phi][x_, y_] = 1 + x\/\[Pi]\ \((ArcTan[\(1 - x\)\/y] - ArcTan[\(\(-1\) - x\)\/y])\)\[IndentingNewLine] \(\(-\(1\/\[Pi]\)\) \((ArcTan[\(-1\) + x, y] + ArcTan[1 + x, y])\) + \(y\/\(2 \[Pi]\)\) Log[\(1 - 2\ x + x\^2 + y\^2\)\/\(1 + 2\ x + x\^2 + y\^2\)];\)\ \), \ "\[IndentingNewLine]", \(\(ContourPlot[\[Phi][x, y], {x, \(-3\), 3}, {y, 0.000001, 3}, \[IndentingNewLine]Contours \[Rule] 10, PlotPoints \[Rule] 35, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]ColorFunction \[Rule] Hue];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nThen use ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a 3D plot of the solution." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Plot3D[\[Phi][x, y], {x, \(-3\), 3}, {y, 10\^\(-6\), 2}, PlotPoints \[Rule] {31, 21}, PlotRange \[Rule] {{\(-3\), 3}, {0, 2}, {\(-1\), 1}}, AxesLabel \[Rule] {"\", "\", "\< \[Phi][x,y]\>"}, Ticks \[Rule] {Range[\(-5\), 5, 1], Range[\(-5\), 5, 0.5], Range[\(-1\), 1, 0.5]}, ViewPoint \[Rule] {\(-2.5\), \(-3\), 2}, ColorFunction \[Rule] Hue];\)\ \), "\n", \(\(Print["\<\[Phi][x,y] = \>", \[Phi][x, y]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nTherefore, the function \n", Cell[BoxData[{ \(\[Phi][x, y] = 1 + x\/\[Pi]\ \((ArcTan[\(1 - x\)\/y] - ArcTan[\(\(-1\) - x\)\/y])\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(-\(1\/\[Pi]\)\) \ \((ArcTan[\(-1\) + x, y] + ArcTan[1 + x, y])\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(+\(y\/\(2 \[Pi]\)\)\) Log[\(1\ - \ 2\ x\ + \ x\^2\ + \ y\^2\)\/\(1\ + \ 2\ x\ + \ x\^2\ \ + \ y\^2\)]\)}]], " \nis harmonic in the upper half-plane ", Cell[BoxData[ \(Im[z] > 0\)]], ", and takes on the desired boundary values. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Library Research Experience for Undergraduates", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.996109]]], "Text"], Cell[TextData[{ StyleBox["Project I. Write on the Dirichlet problem and include some \ applications.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0, 0.996109]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Flanigan, Francis J., (1973), ''Some Half-Plane Dirichlet Problems: A \ Bare Hands Approach (in Classroom Notes),'' Am. Math. M., Vol. 80, No. 1. \ (Jan., 1973), pp. 59-61, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["2.", FontWeight->"Bold"], " Garding, Lars, (1979), ''The Dirichlet Problem,'' Math. Intell., V. 2, \ No. 1, pp. 43-53.\n\n", StyleBox["3.", FontWeight->"Bold"], " Goulet, John, (1983), ''The Dirichlet Problem: A Mathematical \ Development,'' Pi Mu Epsilon J., V. 7, No. 8, pp. 502-511.\n\n", StyleBox["4.", FontWeight->"Bold"], " Minda, David, ''The Dirichlet Problem for a Disk (in Notes),'' Am. Math. \ M., Vol. 97, No. 3. (Mar., 1990), pp. 220-223, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["5.", FontWeight->"Bold"], " Netuka, Ivan, ''The Dirichlet Problem for Harmonic Functions,'' Am. Math. \ M., Vol. 87, No. 8. (Oct., 1980), pp. 621-628, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " " }], "Text"] }, Closed]], Cell[TextData[{ StyleBox["Section 10.3", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0.250004, 0.250004]], StyleBox[" ", FontSize->16, FontWeight->"Bold"], StyleBox["Exercises for Poisson's Integral for the Upper Half-Plane\n", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0.500008, 0.996109]], StyleBox["See textbook page 405.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]] }], "Text"] }, Open ]], Cell[TextData[{ StyleBox["Section 10", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox[".4", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox["\t", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 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FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox["\t", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[ButtonBox["Image of a Fluid Flow", ButtonData:>{"ca1010.nb", None}, ButtonStyle->"Hyperlink"], FontSize->14, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["Section 10", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox[".11", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox["\t", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[ButtonBox["Sources and Sinks", ButtonData:>{"ca1011.nb", None}, ButtonStyle->"Hyperlink"], FontSize->14, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["GO TO THE", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[0, 0.996109, 0]], StyleBox[" ", FontSize->18, FontWeight->"Bold"], 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Make modifications to any definition using \ commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, ScriptMinSize->12, FontSize->16], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellBracketOptions->{"Margins"->{1, 1}, "Widths"->{0, 5}}, ScriptMinSize->8, FontSize->11], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ScriptMinSize->5, FontSize->10, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{12, Inherited}, {20, 40}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->36, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {15, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {10, 15}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, 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Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {8, 10}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, CounterIncrements->"Subsubsection", FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text"], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {6, 6}}, LineSpacing->{1, 3}, CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontSize->9] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, 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CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, ScriptLevel->0, SingleLetterItalics->True, StyleMenuListing->None, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ 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FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->16, FontColor->RGBColor[1, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. 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