(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 68926, 2222] NotebookOptionsPosition[ 53715, 1854] NotebookOutlinePosition[ 64096, 2089] CellTagsIndexPosition[ 64013, 2084] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.4910584115*^9, 3.4910584118125*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Calculating images under complex functions", "Subtitle", ShowCellTags->False, CellChangeTimes->{{3.494447298796875*^9, 3.494447320609375*^9}}, TextAlignment->Center, TextJustification->0], Cell["25 September 2010", "Subsubtitle", CellChangeTimes->{{3.48893028171875*^9, 3.48893030315625*^9}, { 3.489583451984375*^9, 3.489583455796875*^9}, 3.49028881771875*^9, { 3.4905515529375*^9, 3.490551553125*^9}, 3.490875430171875*^9, { 3.492293113328125*^9, 3.492293120375*^9}, 3.49346163509375*^9, { 3.49410698178125*^9, 3.494106982265625*^9}, 3.494416320359375*^9}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2010 by Murray Eisenberg. All rights reserved.\ \>", "SmallText", ShowCellTags->False, CellChangeTimes->{{3.4910584066875*^9, 3.49105843275*^9}, 3.49444721809375*^9}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.49444756890625*^9, 3.494447570546875*^9}}], Cell[TextData[{ "Notebook ", StyleBox["VisualizingFunctions.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " shows how to visualize how a function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]]], " maps the complex plane by drawing geometric objects\[LongDash]including \ line segments, rays, circles, and Cartesian and polar grids\[LongDash]and \ their images under the function." }], "Text", CellChangeTimes->{{3.494447409359375*^9, 3.494447541*^9}, { 3.49450020871875*^9, 3.494500209140625*^9}}], Cell["\<\ Such drawings suggest descriptions of how such a function maps the plane. But \ a drawing can only suggest how the function maps, definitely prove it. To \ prove the way a function maps, calculations are required.\ \>", "Text", CellChangeTimes->{{3.49444754359375*^9, 3.494447744171875*^9}}], Cell["\<\ The purpose of the present notebook is to illustrate such calculations.\ \>", "Text", CellChangeTimes->{{3.49444774975*^9, 3.494447762015625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Prerequisites", "Section", CellChangeTimes->{{3.466616366328125*^9, 3.46661637828125*^9}, { 3.466616604453125*^9, 3.466616607390625*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematica", FontSlant->"Italic"]], "Subsection", CellChangeTimes->{{3.466616614109375*^9, 3.466616621015625*^9}}], Cell[TextData[{ "This notebook does ", StyleBox["not", FontSlant->"Italic"], " use ", StyleBox["Presentations", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.46661644090625*^9, 3.466616483484375*^9}, { 3.466616542578125*^9, 3.466616591375*^9}, {3.4666167129375*^9, 3.466616714296875*^9}, {3.490983043921875*^9, 3.490983044875*^9}, { 3.490984276765625*^9, 3.4909843116875*^9}, 3.4910591349375*^9, { 3.494447877734375*^9, 3.4944478955*^9}, {3.49450865565625*^9, 3.4945086715*^9}}], Cell[TextData[{ "This notebook serves as a companion to ", StyleBox["VisualizingPresentations.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Whereas the latter notebook shows how to visualize the images of \ geometric objects such as points, lines, rays, and circular arcs under a \ function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]]], ", the present notebook shows how to calculate such images symbolically." }], "Text", CellChangeTimes->{{3.4945004325625*^9, 3.494500535671875*^9}, { 3.49450089715625*^9, 3.494500902390625*^9}}], Cell[TextData[{ "This notebook also includes, for convenience, an extract from ", StyleBox["VisualizingPresentations.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " about the Cartesian form of a complex function." }], "Text", CellChangeTimes->{{3.494501052234375*^9, 3.49450115640625*^9}, { 3.494514075828125*^9, 3.494514091609375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{{3.46661662978125*^9, 3.466616631109375*^9}}], Cell[TextData[{ "You should already know the algebra of complex numbers and polar \ representation of complex numbers as well as the idea of a function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]]], ", that is, a complex-valued function of a complex variable." }], "Text", CellChangeTimes->{{3.466616634859375*^9, 3.46661670978125*^9}, { 3.49105958428125*^9, 3.4910595870625*^9}, {3.49105962053125*^9, 3.491059636984375*^9}, {3.4910599269375*^9, 3.4910599291875*^9}, { 3.4924712228125*^9, 3.492471275453125*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Example: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", SuperscriptBox["z", "2"]}], TraditionalForm]], "None", FormatType->"TraditionalForm"] }], "Section", CellChangeTimes->{{3.49444777128125*^9, 3.4944477821875*^9}}], Cell[TextData[{ "As in notebook ", StyleBox["VisualizingFunctions.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", the principal example here will be the squaring function." }], "Text", CellChangeTimes->{{3.494447789578125*^9, 3.494447809609375*^9}, { 3.494500404140625*^9, 3.4945004080625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", CellChangeTimes->{{3.494447811640625*^9, 3.49444781484375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Images in Cartesian coordinates", "Section", CellChangeTimes->{{3.49450101671875*^9, 3.494501023375*^9}}], Cell[CellGroupData[{ Cell["Expressing a function in Cartesian form", "Subsection", CellChangeTimes->{{3.492607300546875*^9, 3.492607317453125*^9}, 3.4944412561875*^9, {3.4945012025*^9, 3.494501217296875*^9}, { 3.49450158440625*^9, 3.494501598109375*^9}}], Cell[TextData[{ "Here is the formula for the function ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " in terms of the real part ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and the imaginary part ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " of an input ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", "=", " ", RowBox[{"x", " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "y"}]}]}], TraditionalForm]]], "\[Ellipsis]" }], "Text", ShowCellTags->False], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.491059938453125*^9, 3.491059939578125*^9}}], Cell["\<\ \[Ellipsis]and here is that same formula in Cartesian form after the squaring \ is done:\ \>", "Text", ShowCellTags->False, CellChangeTimes->{{3.4910599746875*^9, 3.491059978796875*^9}, { 3.492471390078125*^9, 3.49247139215625*^9}, 3.492520122515625*^9, { 3.494501666515625*^9, 3.49450166715625*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{ 3.491059950515625*^9, {3.491059982328125*^9, 3.49106002703125*^9}, { 3.492471402578125*^9, 3.492471414828125*^9}, {3.494501231625*^9, 3.494501234390625*^9}}], Cell[TextData[{ "(", StyleBox["Expand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " will give you the same thing there as ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".)" }], "Text", CellChangeTimes->{{3.492471430296875*^9, 3.49247145034375*^9}, { 3.4945012485*^9, 3.49450125665625*^9}, 3.494518559953125*^9}], Cell[TextData[{ "Form the ", StyleBox["real part", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and", StyleBox[" imaginary part", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " of the function ", Cell[BoxData[ FormBox["f", TraditionalForm]]], ". 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In ", StyleBox["Mathematica:", FontSlant->"Italic"] }], "Text", ShowCellTags->False, CellChangeTimes->{{3.491060078734375*^9, 3.491060106171875*^9}, 3.492471503109375*^9, {3.49450134315625*^9, 3.494501345953125*^9}, { 3.494501503625*^9, 3.494501533046875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{ RowBox[{ RowBox[{"u", "[", RowBox[{"x_", ",", "y_"}], "]"}], " ", ":=", " ", RowBox[{"ComplexExpand", "@", RowBox[{"Re", "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v", "[", RowBox[{"x_", ",", "y_"}], "]"}], " ", ":=", RowBox[{"ComplexExpand", "@", RowBox[{"Im", "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]}]}]}], "Input", ShowCellTags->False, CellChangeTimes->{{3.49106011096875*^9, 3.49106017865625*^9}, 3.49106062825*^9, {3.4924715149375*^9, 3.49247151640625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "y"}], 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In other words, what is an equation relating ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], "? We prefer to be able to express one of ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " as a function of the other. Or, if that is not possible, at least to find \ an implicit relationship between ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.494448131140625*^9, 3.494448279546875*^9}, { 3.494501813625*^9, 3.494501816*^9}}], Cell[TextData[{ "The real and imaginary parts ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], " of ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "z", ")"}], TraditionalForm]]], " are the entries in the list:" }], "Text", CellChangeTimes->{{3.494448074828125*^9, 3.49444809925*^9}, { 3.494501837265625*^9, 3.494501845453125*^9}}], Cell[BoxData[ RowBox[{"uv", "=", RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "@", RowBox[{"f", "[", "z", "]"}]}], ",", RowBox[{"Im", "@", RowBox[{"f", "[", "z", "]"}]}]}], "}"}], "]"}]}]], "Input", CellChangeTimes->{{3.49444803278125*^9, 3.4944480705*^9}, { 3.4944481030625*^9, 3.49444811003125*^9}, {3.494500674890625*^9, 3.49450069284375*^9}}], Cell[TextData[{ "Then the equations relating ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], " to ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " are given by:" }], "Text", CellChangeTimes->{{3.49450754634375*^9, 3.49450756840625*^9}}], Cell[BoxData[ RowBox[{"Thread", "[", RowBox[{ RowBox[{"{", RowBox[{"u", ",", "v"}], "}"}], "\[Equal]", "uv"}], "]"}]], "Input", CellChangeTimes->{{3.494507575828125*^9, 3.494507583265625*^9}}], Cell[TextData[{ "To obtain a relationship between ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], ", eliminate the parameter ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " for the line from the equations:\n", "\t", Cell[BoxData[ RowBox[{ StyleBox["{", FontFamily->"Times"], GridBox[{ {Cell[TextData[StyleBox["u", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"]]], StyleBox["=", FontFamily->"Times"], RowBox[{ RowBox[{ SuperscriptBox[ StyleBox["x", FontFamily->"Times", FontSlant->"Italic"], "2"], StyleBox["-", FontFamily->"Times"], SuperscriptBox[ StyleBox["b", FontFamily->"Times", FontSlant->"Italic"], "2"]}], StyleBox[",", FontFamily->"Times"]}]}, { StyleBox["v", FontFamily->"Times", FontSlant->"Italic"], StyleBox["=", FontFamily->"Times"], RowBox[{"2", StyleBox["b", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Times", FontSlant->"Italic"], StyleBox[ RowBox[{"x", "."}], FontFamily->"Times", FontSlant->"Italic"]}]} }]}]]], " \t(*)" }], "Text", CellChangeTimes->{{3.494448297453125*^9, 3.494448384859375*^9}, { 3.494448548609375*^9, 3.494448571984375*^9}, {3.494501885703125*^9, 3.494501896421875*^9}, {3.49450761459375*^9, 3.494507636703125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "To do that, first solve the second equation for ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " as a function of ", Cell[BoxData[ FormBox["v", TraditionalForm]]], ":" }], "Text", CellChangeTimes->{{3.4944486085625*^9, 3.494448622171875*^9}, { 3.49450190878125*^9, 3.49450190975*^9}}], Cell[BoxData[ RowBox[{"xValue", "=", RowBox[{"x", "/.", RowBox[{"First", "@", RowBox[{"Solve", "[", RowBox[{ RowBox[{"v", "\[Equal]", RowBox[{"Last", "@", "uv"}]}], ",", "x"}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.49444857440625*^9, 3.4944485923125*^9}, { 3.4945062675625*^9, 3.494506275515625*^9}}], Cell[TextData[{ "Of course that makes no sense when ", Cell[BoxData[ FormBox[ RowBox[{"b", "=", "0"}], TraditionalForm]]], ", so consider two cases." }], "Text", CellChangeTimes->{{3.4944486264375*^9, 3.4944486576875*^9}, { 3.494460557640625*^9, 3.4944605751875*^9}, {3.494501919578125*^9, 3.494501921296875*^9}}], Cell[TextData[{ StyleBox["Case (i):", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox[ RowBox[{"b", "\[NotEqual]", "0"}], TraditionalForm]]], ". Then the preceding expression for ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " as a function of ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " makes sense. Substitute that expression for ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " into the first equation:" }], "Text", CellChangeTimes->{{3.4944486264375*^9, 3.4944486576875*^9}, { 3.494460557640625*^9, 3.4944606364375*^9}, {3.49450193*^9, 3.494501932921875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"u", "==", RowBox[{"First", "@", "uv"}]}], "/.", RowBox[{"x", "\[Rule]", "xValue"}]}]], "Input", CellChangeTimes->{{3.494448667640625*^9, 3.49444868753125*^9}, 3.49444872003125*^9, {3.494506279421875*^9, 3.4945062830625*^9}}], Cell[TextData[{ "That equation expresses ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], " explicitly as a function of ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], ". 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In this case, the real and imaginary parts of ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "z", ")"}], TraditionalForm]]], " are given by: " }], "Text", CellChangeTimes->{{3.494448694765625*^9, 3.494448755046875*^9}, { 3.494460646671875*^9, 3.494460785640625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "@", RowBox[{"f", "[", "z", "]"}]}], ",", RowBox[{"Im", "@", RowBox[{"f", "[", "z", "]"}]}]}], "}"}], "]"}], "/.", RowBox[{"b", "\[Rule]", "0"}]}]], "Input", CellChangeTimes->{{3.49444803278125*^9, 3.4944480705*^9}, { 3.4944481030625*^9, 3.49444811003125*^9}, {3.49446075425*^9, 3.494460756859375*^9}, 3.494461475453125*^9}], Cell[TextData[{ "In other words:\n\t", Cell[BoxData[ RowBox[{ StyleBox["{", FontFamily->"Times"], GridBox[{ { StyleBox["u", FontFamily->"Times", FontSlant->"Italic"], StyleBox["=", FontFamily->"Times"], RowBox[{ SuperscriptBox[ StyleBox["x", FontFamily->"Times", FontSlant->"Italic"], StyleBox["2", FontFamily->"Times"]], StyleBox[",", FontFamily->"Times"]}]}, { StyleBox["v", FontFamily->"Times", FontSlant->"Italic"], StyleBox["=", FontFamily->"Times"], StyleBox["0.", FontFamily->"Times"]} }]}]]], "\t(**)" }], "Text", CellChangeTimes->{{3.494501959453125*^9, 3.494502001140625*^9}, { 3.49450303421875*^9, 3.494503036515625*^9}}], Cell[TextData[{ "Thus in this case, the image lies on the graph of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "\[GreaterEqual]", "0"}], ",", " ", RowBox[{"v", "=", "0"}]}], TraditionalForm]]], ", which is the non-negative real-axis." }], "Text", CellChangeTimes->{{3.4944607936875*^9, 3.49446084884375*^9}, 3.494518570734375*^9}], Cell[TextData[{ StyleBox["The reasoning so far is incomplete!", FontSlant->"Italic"], " All it establishes is that ", StyleBox["if", FontWeight->"Bold", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " lies on the horizontal line ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Im", "(", "z", ")"}], "=", "b"}], TraditionalForm]]], ", ", StyleBox["then", FontWeight->"Bold", FontSlant->"Italic"], " its image ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "z", ")"}], TraditionalForm]]], " lies on the parabola ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "+", SuperscriptBox["b", "2"]}], "=", RowBox[{ RowBox[{"(", RowBox[{"1", "/", RowBox[{"(", RowBox[{"4", SuperscriptBox["b", "2"]}], ")"}]}], ")"}], SuperscriptBox["v", "2"]}]}], TraditionalForm]]], " in Case (", StyleBox["i", FontSlant->"Italic"], "), and on the non-negative real-axis ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "\[GreaterEqual]", "0"}], ",", " ", RowBox[{"v", "=", "0"}]}], TraditionalForm]]], " in Case (", StyleBox["ii", FontSlant->"Italic"], "). In other words, the image of the horizontal line is a subset of the \ parabola or the half-axis, as the case may be." }], "Text", CellChangeTimes->{{3.49446086465625*^9, 3.494461097390625*^9}, { 3.49446120965625*^9, 3.494461231859375*^9}, {3.494502362765625*^9, 3.494502363265625*^9}, {3.494503269953125*^9, 3.494503269953125*^9}, 3.494518573859375*^9}], Cell[TextData[{ StyleBox["\[WarningSign]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0.8]], StyleBox[" ", FontWeight->"Bold", Background->RGBColor[1, 1, 0.8]], StyleBox["Caution", FontWeight->"Bold", FontSlant->"Italic", Background->RGBColor[1, 1, 0.8]], StyleBox[":", FontWeight->"Bold", Background->RGBColor[1, 1, 0.8]], " The reasoning so far does ", StyleBox["not", FontSlant->"Italic"], " establish the reverse inclusion\[LongDash]that such a parabola, or the \ half-axis, is a subset of the image of the line in one case or the other. In \ other words, it does ", StyleBox["not", FontSlant->"Italic"], " establish that ", StyleBox["if", FontWeight->"Bold", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["w", TraditionalForm]]], " is a point on the parabola, or on the half-axis, ", StyleBox["then", FontWeight->"Bold", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["w", TraditionalForm]]], " is the value of ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " for some ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " on the original horizontal line!" }], "Text", CellChangeTimes->{{3.49446086465625*^9, 3.494461197671875*^9}, { 3.494461250484375*^9, 3.494461272734375*^9}, {3.49450209640625*^9, 3.4945021155625*^9}, 3.4945026494375*^9, {3.49450268153125*^9, 3.49450268403125*^9}, 3.494502763875*^9, 3.494518581875*^9}], Cell["\<\ To complete the reasoning, again consider the two cases separately.\ \>", "Text", CellChangeTimes->{{3.494448694765625*^9, 3.494448800515625*^9}, { 3.494461281359375*^9, 3.494461293265625*^9}}], Cell[TextData[{ StyleBox["Case (i):", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox[ RowBox[{"b", "\[NotEqual]", "0"}], TraditionalForm]]], ". 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Thus ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}]}], TraditionalForm]]], " with ", Cell[BoxData[ FormBox[ RowBox[{"u", "\[GreaterEqual]", "0"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.494448694765625*^9, 3.494448755046875*^9}, { 3.494460646671875*^9, 3.494460785640625*^9}, 3.494461322859375*^9, { 3.494461371078125*^9, 3.4944613729375*^9}, {3.494503002359375*^9, 3.494503023640625*^9}, 3.49450305978125*^9, {3.494506810828125*^9, 3.494506823734375*^9}, 3.49451858915625*^9}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"x", ",", "y", ",", "u", ",", "v", ",", "w"}], "]"}]], "Input", CellChangeTimes->{{3.494506506921875*^9, 3.494506511203125*^9}, { 3.494506833484375*^9, 3.494506834046875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"w", "=", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.494506838984375*^9, 3.4945068453125*^9}}], Cell[TextData[{ "When ", Cell[BoxData[ FormBox[ RowBox[{"b", "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ", the value of ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " at ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "b"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " is:" }], "Text", CellChangeTimes->{{3.494506532546875*^9, 3.49450656515625*^9}}], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.49450650340625*^9, 3.49450652421875*^9}, { 3.4945065721875*^9, 3.4945065831875*^9}, {3.494506858796875*^9, 3.494506860265625*^9}}], Cell[TextData[{ "We seek ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " with ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", that is, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{ SuperscriptBox["x", "2"], "=", "u"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". " }], "Text", CellChangeTimes->{{3.494506587578125*^9, 3.49450667828125*^9}, 3.49450688096875*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"x", "=", RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}], "]"}], "\[Equal]", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}]}], ",", "x"}], "]"}]}]}], ";"}], "\n", RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "0"}]}]}]}], "Input", CellChangeTimes->{{3.494506681078125*^9, 3.4945067833125*^9}, { 3.49450686646875*^9, 3.494506871140625*^9}, {3.494506910421875*^9, 3.494506914734375*^9}, {3.4945069464375*^9, 3.4945069619375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z", "]"}], "\[Equal]", RowBox[{"{", RowBox[{"w", ",", "w"}], "}"}]}]], "Input", CellChangeTimes->{{3.4945069760625*^9, 3.49450699971875*^9}}], Cell[TextData[{ "That completes the reasoning in Case (", StyleBox["ii", FontSlant->"Italic"], ")." }], "Text", CellChangeTimes->{{3.494507012734375*^9, 3.4945070240625*^9}}], Cell[TextData[{ StyleBox["Note.", FontWeight->"Bold"], " For this example, it is probably simpler to do all the calculations with \ paper and pencil than to use ", StyleBox["Mathematica", FontSlant->"Italic"], " as an aid. With a more complicated function, you may want the assistance \ of ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.494507993328125*^9, 3.494508053625*^9}, { 3.494513758703125*^9, 3.494513858359375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Similarly, determine the image of a vertical line ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Re", "(", "z", ")"}], "=", "a"}], TraditionalForm]], FormatType->"TraditionalForm"], " under ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", SuperscriptBox["z", "2"]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{{3.494507872265625*^9, 3.49450794565625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Determine the image of an arbitrary horizontal line and of an arbitrary \ vertical line under ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"h", "(", "z", ")"}], "=", RowBox[{"z", "\[Conjugate]"}]}], TraditionalForm]]], ", the conjugate." }], "Exercise", CellChangeTimes->{{3.494507872265625*^9, 3.49450794565625*^9}, { 3.49451387178125*^9, 3.49451392921875*^9}, {3.494514326453125*^9, 3.494514329015625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Repeat the preceding exercise but for the function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"k", "(", "z", ")"}], "=", RowBox[{"1", "/", "z"}]}], TraditionalForm]]], ". (Of course, exclude the origin from the domain in the case of a \ horizontal or vertical line going through the origin, that is in the case of \ the line being the real or imaginary axis.)" }], "Exercise", CellChangeTimes->{{3.494507872265625*^9, 3.49450794565625*^9}, { 3.49451387178125*^9, 3.49451402409375*^9}, 3.494514321453125*^9, 3.49451860740625*^9}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Images in polar coordinates", "Section", CellChangeTimes->{{3.49450101671875*^9, 3.494501023375*^9}, { 3.494501559828125*^9, 3.494501560453125*^9}}], Cell["Recall that we are using the complex function:", "Text", CellChangeTimes->{{3.494501760140625*^9, 3.494501768640625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", CellChangeTimes->{{3.49450179028125*^9, 3.49450179415625*^9}}], Cell[TextData[{ StyleBox["Problem:", FontWeight->"Bold"], " What is the image under ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " of a ray ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Arg", "(", "z", ")"}], "=", "\[Alpha]"}], TraditionalForm]]], " emanating from the origin?" }], "Text", CellChangeTimes->{{3.49444792790625*^9, 3.49444797984375*^9}, { 3.49450147921875*^9, 3.494501482234375*^9}, {3.494501726984375*^9, 3.494501727953125*^9}, {3.49450828171875*^9, 3.49450829875*^9}, { 3.494512827359375*^9, 3.49451283646875*^9}}], Cell[TextData[{ "Actually, that description of the ray is misleading, since the origin 0 is \ not in the domain of ", Cell[BoxData[ FormBox["Arg", TraditionalForm]], FormatType->"TraditionalForm"], ". However, 0 is on every ray emanating from the origin, and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "0", ")"}], "=", RowBox[{ SuperscriptBox["0", "2"], "=", "0"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " is also on every ray emanating from the origin." }], "Text", CellChangeTimes->{{3.494512775296875*^9, 3.494512909859375*^9}}], Cell[TextData[{ "From now on, consider only ", StyleBox["nonzero", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49451291209375*^9, 3.494512924140625*^9}}], Cell["\<\ Clear any assignments to the variables we are going to use in the domain and \ codomain planes:\ \>", "Text", CellChangeTimes->{{3.4945007139375*^9, 3.494500755515625*^9}, { 3.49450172003125*^9, 3.4945017450625*^9}, 3.494512934515625*^9}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"\[Alpha]", ",", "z", ",", "r", ",", "w", ",", "s"}], "]"}]], "Input", CellChangeTimes->{{3.494500731796875*^9, 3.494500736421875*^9}, { 3.4945083169375*^9, 3.49450836340625*^9}, {3.49450927771875*^9, 3.49450927915625*^9}, {3.4945130599375*^9, 3.49451306903125*^9}, 3.494518061796875*^9, 3.49451837584375*^9}], Cell[TextData[{ "An arbitrary point ", Cell[BoxData[ FormBox[ RowBox[{"z", "\[NotEqual]", "0"}], TraditionalForm]]], " on the ray has the following form, for some ", Cell[BoxData[ FormBox[ RowBox[{"r", ">", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ":" }], "Text", CellChangeTimes->{{3.494508374796875*^9, 3.4945083835625*^9}, { 3.494512944734375*^9, 3.494512978*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"z", "=", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Alpha]"}], "]"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.49450838509375*^9, 3.4945083963125*^9}, 3.494512958921875*^9}], Cell[TextData[{ "Its image under ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " is:" }], "Text", CellChangeTimes->{{3.49450840240625*^9, 3.494508406796875*^9}}], Cell[BoxData[ RowBox[{"f", "[", "z", "]"}]], "Input", CellChangeTimes->{{3.49450840790625*^9, 3.494508411671875*^9}}], Cell[TextData[{ "Hence ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "z", ")"}], TraditionalForm]]], " is a nonzero point on the ray consisting of all points ", Cell[BoxData[ FormBox["w", TraditionalForm]]], " such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", "\[Alpha]"}], "\[Element]", RowBox[{"arg", "(", "w", ")"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.49450849125*^9, 3.494508587109375*^9}, 3.49450920859375*^9, {3.49451303721875*^9, 3.494513039015625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Immediately above, why did we not say \"the ray with equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Arg", "(", "w", ")"}], "=", RowBox[{"2", "\[Alpha]"}]}], TraditionalForm]]], "?" }], "Exercise", CellChangeTimes->{{3.49450859453125*^9, 3.494508632890625*^9}, 3.49450874009375*^9}], Cell[TextData[{ StyleBox["The reasoning so far is incomplete!", FontSlant->"Italic"], " All it establishes is that ", StyleBox["if", FontWeight->"Bold", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " lies on the ray Arg(z) = \[Alpha], ", StyleBox["then", FontWeight->"Bold", FontSlant->"Italic"], " its image ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "z", ")"}], TraditionalForm]]], " lies on the ray ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"{", RowBox[{"w", "\[Element]", RowBox[{"\[DoubleStruckCapitalC]", ":", RowBox[{ RowBox[{"2", "\[Alpha]"}], "\[Element]", RowBox[{"arg", "(", "w", ")"}]}]}]}], "}"}], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.49446086465625*^9, 3.494461097390625*^9}, { 3.49446120965625*^9, 3.494461231859375*^9}, {3.494502362765625*^9, 3.494502363265625*^9}, {3.494503269953125*^9, 3.494503269953125*^9}, { 3.494508875765625*^9, 3.494508962796875*^9}, 3.4945092174375*^9}], Cell[TextData[{ StyleBox["\[WarningSign]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0.8]], StyleBox[" ", FontWeight->"Bold", Background->RGBColor[1, 1, 0.8]], StyleBox["Caution", FontWeight->"Bold", FontSlant->"Italic", Background->RGBColor[1, 1, 0.8]], StyleBox[":", FontWeight->"Bold", Background->RGBColor[1, 1, 0.8]], " The reasoning so far does ", StyleBox["not", FontSlant->"Italic"], " establish the reverse inclusion\[LongDash]that the entire latter ray is a \ subset of the image of the original ray. In other words, it does ", StyleBox["not", FontSlant->"Italic"], " establish that ", StyleBox["if", FontWeight->"Bold", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["w", TraditionalForm]]], " is a point with ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", "\[Alpha]"}], "\[Element]", RowBox[{"arg", "(", "w", ")"}]}], TraditionalForm]]], ", ", StyleBox["then", FontWeight->"Bold", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["w", TraditionalForm]]], " is the value of ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " at some ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " on the original ray!" }], "Text", CellChangeTimes->{{3.49446086465625*^9, 3.494461197671875*^9}, { 3.494461250484375*^9, 3.494461272734375*^9}, {3.49450209640625*^9, 3.4945021155625*^9}, 3.4945026494375*^9, {3.49450268153125*^9, 3.49450268403125*^9}, 3.494502763875*^9, {3.49450897390625*^9, 3.494509036359375*^9}, 3.494509227921875*^9}], Cell[TextData[{ "To complete the reasoning, let ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{"s", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Phi]"}]]}]}], TraditionalForm]]], " be a nonzero point on the ray given by ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", "\[Alpha]"}], "\[Element]", RowBox[{"arg", "(", "w", ")"}]}], TraditionalForm]]], ". In other words:" }], "Text", CellChangeTimes->{{3.494509039109375*^9, 3.4945090435*^9}, { 3.494509076578125*^9, 3.49450920153125*^9}, {3.494509238203125*^9, 3.494509254640625*^9}, {3.494513044578125*^9, 3.4945130455625*^9}}], Cell[BoxData[ RowBox[{"w", "=", RowBox[{"s", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "2", "\[Alpha]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.494509414125*^9, 3.49450942190625*^9}, { 3.49451804934375*^9, 3.49451805053125*^9}}], Cell[TextData[{ "The goal is to find some nonzero ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], " with ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", "w"}], TraditionalForm]], FormatType->"TraditionalForm"], ". This equation takes the form:" }], "Text", CellChangeTimes->{{3.49451328703125*^9, 3.494513308890625*^9}, { 3.494514531796875*^9, 3.494514533859375*^9}}], Cell[BoxData[ RowBox[{"polarEqn", "=", RowBox[{ RowBox[{"f", "[", "z", "]"}], "\[Equal]", "w"}]}]], "Input", CellChangeTimes->{{3.494513238875*^9, 3.4945132650625*^9}, { 3.494513337640625*^9, 3.49451333834375*^9}, {3.4945183536875*^9, 3.494518355375*^9}}], Cell[TextData[{ "That is equivalent to the moduli of the two sides being equal. To say that \ in ", StyleBox["Mathematica", FontSlant->"Italic"], " without copying or re-typing parts of the equation above is not as \ straightforward as you might think. You must tell ", StyleBox["Mathematica", FontSlant->"Italic"], " explicitly that ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\[Phi]", TraditionalForm]]], " are real. And you may do that by using the rules that their imaginary \ parts are 0:" }], "Text", CellChangeTimes->{{3.49451365878125*^9, 3.49451368246875*^9}, { 3.4945145400625*^9, 3.4945146289375*^9}, {3.494518101984375*^9, 3.494518173671875*^9}}], Cell[BoxData[ RowBox[{"absPolarEqn", "=", RowBox[{ RowBox[{"Abs", "/@", "polarEqn"}], "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Im", "[", "\[Alpha]", "]"}], "\[Rule]", "0"}], ",", RowBox[{ RowBox[{"Im", "[", "\[Phi]", "]"}], "\[Rule]", "0"}]}], "}"}]}]}]], "Input", CellChangeTimes->{{3.494513641140625*^9, 3.494513647828125*^9}, { 3.494514714703125*^9, 3.494514742703125*^9}, {3.49451840275*^9, 3.4945184078125*^9}}], Cell[TextData[{ "Because ", Cell[BoxData[ FormBox[ RowBox[{"w", "\[NotEqual]", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ", then ", Cell[BoxData[ FormBox[ RowBox[{"s", ">", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ". 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