(* 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[ 77551, 2324] NotebookOptionsPosition[ 61575, 1925] NotebookOutlinePosition[ 72136, 2166] CellTagsIndexPosition[ 72022, 2160] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.49777880134375*^9, 3.49777880184375*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Complex sine function", "Subtitle", TextAlignment->Center, TextJustification->0], Cell["7 November 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, { 3.49777882415625*^9, 3.49777882675*^9}, 3.498071466125*^9, 3.49815800878125*^9}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2004\[Dash]2010 by Murray Eisenberg. All rights \ reserved.\ \>", "SmallText", CellChangeTimes->{{3.497778805671875*^9, 3.497778806046875*^9}, 3.49777895390625*^9}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.4978090355*^9, 3.49780903709375*^9}}], Cell["\<\ This notebook examines symbolically and, especially, graphically, fundamental \ properties of the complex sine function and the complex inverse sine function.\ \>", "Text", CellChangeTimes->{{3.49780903915625*^9, 3.4978091*^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[{ "Much of this notebook requires David Park's ", StyleBox["Mathematica", FontSlant->"Italic"], " add-on application ", 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, { 3.497809121734375*^9, 3.49780912225*^9}}], Cell[TextData[{ StyleBox["Presentations", FontSlant->"Italic"], " should be loaded by evaluating the expression:\n\t", StyleBox["<<", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["Presentations", FontFamily->"Courier", FontWeight->"Plain"], StyleBox["`", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nThat initialization is done below, in the ", ButtonBox["Initialization section", BaseStyle->"Hyperlink", ButtonData->"initialization"], ", below." }], "Text", CellChangeTimes->{{3.46661644090625*^9, 3.466616483484375*^9}, { 3.466616542578125*^9, 3.466616591375*^9}, {3.4666167129375*^9, 3.46661679353125*^9}, {3.4666168744375*^9, 3.466616901171875*^9}, { 3.4909830825*^9, 3.4909830910625*^9}, {3.490984498375*^9, 3.49098451746875*^9}, {3.491058732328125*^9, 3.491058740734375*^9}, 3.491058892734375*^9, {3.491058930921875*^9, 3.491059018609375*^9}, { 3.491059057390625*^9, 3.491059075953125*^9}, {3.49105918140625*^9, 3.49105918140625*^9}, {3.491059247640625*^9, 3.49105924765625*^9}, { 3.491059412859375*^9, 3.491059447375*^9}, {3.491059659734375*^9, 3.491059692453125*^9}, {3.49105976584375*^9, 3.491059767484375*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "Although the notebook is largely self-contained, it is helpful if you \ already know about using ", StyleBox["Presentations", FontSlant->"Italic"], " to draw a \"two-panel plot\" of the domain and codomain of a complex \ function and to lift objects in the complex plane to the Riemann sphere. See \ notebooks ", StyleBox["VisualizingFunctions.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["RiemannSphere.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". For handling the complex exponential function, see notebook ", StyleBox["ExponentialLogFunctions.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.4910597709375*^9, 3.4910599115*^9}, { 3.49247129434375*^9, 3.492471300390625*^9}, {3.49477440659375*^9, 3.4947744791875*^9}, {3.497809157328125*^9, 3.49780931040625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{{3.46661662978125*^9, 3.466616631109375*^9}}], Cell[TextData[{ "You should already know about the complex exponential function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"exp", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]]], ". See notebook ", StyleBox["ExponentialLogFunctions.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "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}, {3.494774509609375*^9, 3.4947745360625*^9}, {3.4978093309375*^9, 3.497809371171875*^9}, { 3.498072128109375*^9, 3.49807214275*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section", ShowGroupOpener->True, CellChangeTimes->{{3.4906339376875*^9, 3.49063394653125*^9}, { 3.490638317984375*^9, 3.49063833975*^9}, {3.490735161125*^9, 3.490735165125*^9}, 3.490984367984375*^9}, CellTags->"initialization"], Cell[TextData[{ "When you opened this notebook, it should have prompted you whether you want \ to evaluate Initialization Cells. 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Then use the Ratio Test to show that cos is entire." }], "Exercise", CellChangeTimes->{{3.497794830459695*^9, 3.49779486300657*^9}, { 3.497794939209695*^9, 3.497794962740945*^9}, {3.497795032897195*^9, 3.497795045490945*^9}, {3.497809614484375*^9, 3.497809688015625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Converting trig functions to and from the complex exponential function\ \>", "Subsection", CellChangeTimes->{{3.497794753490945*^9, 3.49779477410032*^9}, { 3.49780971075*^9, 3.49780971296875*^9}}], Cell[TextData[{ "The key ", StyleBox["Mathematica", FontSlant->"Italic"], " function to use to express trig and hyperbolic functions in terms of the \ complex exponential is the built-in ", StyleBox["TrigToExp", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ":" }], "Text"], Cell[BoxData[ RowBox[{"?", "TrigToExp"}]], "Input"], Cell["\<\ The complex sine function can be expressed in terms of the complex \ exponential function:\ \>", "Text", CellChangeTimes->{{3.49779506560032*^9, 3.497795089897195*^9}}], Cell[BoxData[ RowBox[{"TrigToExp", "[", RowBox[{"Sin", "[", "z", "]"}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Express the complex cosine and tangent functions in terms of the complex \ exponential function." }], "Exercise", CellChangeTimes->{{3.497794830459695*^9, 3.49779486300657*^9}, { 3.497794939209695*^9, 3.497794962740945*^9}, {3.497795032897195*^9, 3.497795045490945*^9}, {3.4981476463125*^9, 3.49814765490625*^9}}], Cell[TextData[{ "To convert back from exponential form to trig and hyperbolic form, you may \ use the built-in function ", StyleBox["ExpToTrig", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". For example:" }], "Text", CellChangeTimes->{3.49777895928125*^9}], Cell[BoxData[ RowBox[{"ExpToTrig", "[", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "z"}], "]"}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " ", StyleBox["Mathematica", FontSlant->"Italic"], " knew the formula ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"sin", "(", "z", ")"}], "=", RowBox[{ RowBox[{ FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "z"}]]}], "-", RowBox[{ FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "z"}]]}]}]}], TraditionalForm]]], " Prove that the formula is in fact correct." }], "Exercise", CellChangeTimes->{{3.497794830459695*^9, 3.49779486300657*^9}, { 3.497794939209695*^9, 3.49779499103782*^9}, {3.49779512088157*^9, 3.497795227522195*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Trig identities", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Proof of the addition formula for the complex ", Cell[BoxData[ FormBox["sin", TraditionalForm]]] }], "Subsection", CellChangeTimes->{{3.49779478491282*^9, 3.497794785397195*^9}, 3.4981580308125*^9}], Cell["\<\ The Addition Formula for the complex sine just extends the corresponding \ formula for the real sine to complex numbers:\ \>", "Text", CellChangeTimes->{{3.498072230671875*^9, 3.498072373125*^9}}], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"sin", "(", RowBox[{ SubscriptBox["z", "1"], "+", SubscriptBox["z", RowBox[{"2", " "}]]}], ")"}], "=", RowBox[{ RowBox[{ RowBox[{"sin", "(", SubscriptBox["z", "1"], ")"}], " ", RowBox[{"cos", "(", SubscriptBox["z", "2"], ")"}]}], "+", RowBox[{ RowBox[{"cos", "(", SubscriptBox["z", "1"], ")"}], " ", RowBox[{"sin", "(", SubscriptBox["z", "2"], ")"}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "EmphasisText", CellChangeTimes->{{3.498072230671875*^9, 3.498072377*^9}}], Cell[TextData[{ "It is much easier to prove this identity for complex numbers than the \ version for reals if you just express the complex sine and cosine in terms of \ complex exponentials. 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That setting means the drawing \ is scaled to 0.4 times the overall width of the cell. Such a relative ", StyleBox["ImageSize", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " setting will be more appropriate when, below, we combine the \ two-dimensional domain drawing with a three-dimensional drawing of the image \ lifted to the Riemann sphere.)" }], "Text", CellChangeTimes->{{3.497801532272195*^9, 3.49780165928782*^9}, { 3.497810009625*^9, 3.49781006484375*^9}}], Cell[TextData[{ "The following also illustrates the periodicity, but with the image lifted \ to the Riemann sphere. 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Why or why not?" }], "Exercise", CellChangeTimes->{{3.497794830459695*^9, 3.49779486300657*^9}, { 3.49779534216282*^9, 3.497795386147195*^9}, {3.49780217153782*^9, 3.497802381959695*^9}}, ScriptSizeMultipliers->{0.85}], Cell[TextData[{ "But an easier way is to express the complex sine function in terms of the \ complex exponential function and use the fact that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"exp", "'"}], "=", "exp"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49780179260032*^9, 3.49780182728782*^9}}], Cell[BoxData[ RowBox[{"TrigToExp", "[", RowBox[{"Sin", "[", "z", "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"TrigToExp", "[", RowBox[{"Sin", "[", "z", "]"}], "]"}], ",", "z"}], "]"}]], "Input", CellChangeTimes->{{3.49780183075657*^9, 3.49780187053782*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"TrigToExp", "[", RowBox[{"Sin", "[", "z", "]"}], "]"}], ",", "z"}], "]"}], "\[Equal]", 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Check that ", StyleBox["Mathematica", FontSlant->"Italic"], " gives the results you obtain." }], "Exercise", CellChangeTimes->{{3.497794830459695*^9, 3.49779486300657*^9}, { 3.497794939209695*^9, 3.49779499103782*^9}, {3.49779512088157*^9, 3.497795227522195*^9}, {3.497810444703125*^9, 3.497810469765625*^9}, { 3.49781073634375*^9, 3.4978107725*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Mapping properties of complex ", Cell[BoxData[ FormBox["sin", TraditionalForm]]] }], "Section"], Cell["\<\ For several drawings below, we'll use the same Cartesian grid:\ \>", "Text", CellChangeTimes->{{3.498155459703125*^9, 3.498155479*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"grid", "=", RowBox[{"{", RowBox[{ RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", "Linen"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "\[Pi]"}], "-", " ", RowBox[{"1.5", "\[ImaginaryI]"}]}], ",", 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