(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 100814, 2913] NotebookOptionsPosition[ 85259, 2526] NotebookOutlinePosition[ 95808, 2766] CellTagsIndexPosition[ 95730, 2761] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.49746131825*^9, 3.4974613191875*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Exponential and Log functions visualized", "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, { 3.497461383890625*^9, 3.49746139278125*^9}, {3.498070412828125*^9, 3.498070414421875*^9}, 3.49814584684375*^9}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2004\[Dash]2010 by Murray Eisenberg. All rights \ reserved.\ \>", "SmallText", CellChangeTimes->{ 3.497461312390625*^9, {3.497461478953125*^9, 3.4974614833125*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.493499537953125*^9, 3.493499540421875*^9}}], Cell[TextData[{ "This notebook shows how to use David Park's ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " add-on application to visualize the exponential and principal log \ functions. It also uses built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functionality to exhibit and verify some basic properties of those \ functions." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.491058451234375*^9, 3.4910584870625*^9}, 3.4910585549375*^9, {3.49279786209375*^9, 3.492797896046875*^9}, { 3.497461501296875*^9, 3.497461502578125*^9}, {3.497461681515625*^9, 3.49746169915625*^9}, {3.498070540703125*^9, 3.4980705769375*^9}, 3.498071043984375*^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. See the notebook ", StyleBox["VisualizingFunctions.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". One drawing shows the Riemann sphere. See the notebook ", StyleBox["RiemannSphere.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " for ", StyleBox["Mathematica", FontSlant->"Italic"], " background about that." }], "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}, { 3.49807109065625*^9, 3.498071109296875*^9}, {3.498146077625*^9, 3.498146134171875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{{3.46661662978125*^9, 3.466616631109375*^9}}], Cell["\<\ You should already understand the notion of a complex-valued function defined \ on the complex plane and you should know basics about power series in a \ complex variable and how the Ratio Test is used to determine the radius of \ convergence of such a power series.\ \>", "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}, { 3.498071144921875*^9, 3.498071282484375*^9}}], Cell[TextData[{ "One figure illustrates the periodicity of the complex exponential function \ by lifting an image to the Riemann sphere. For information about the Riemann \ sphere and lifting objects from the complex plane to it, see notebook ", StyleBox["RiemannSphere.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.498146144734375*^9, 3.498146189875*^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. You should have answered \ \[OpenCurlyDoubleQuote]yes.\[CloseCurlyDoubleQuote]\nIf you did not, then ", StyleBox["evaluate the following Input cell now", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ". 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Thus:" }], "Text", CellChangeTimes->{{3.497701834578125*^9, 3.497701872203125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Exp", "[", "x", "]"}], "\[Equal]", SuperscriptBox["\[ExponentialE]", "x"]}], "//", "ComplexExpand"}]], "Input", CellChangeTimes->{{3.497701875609375*^9, 3.4977018886875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Derivative of exp", "Subsection", CellChangeTimes->{{3.497700421265625*^9, 3.49770042875*^9}}], Cell[TextData[{ "Since the power series representation of ", Cell[BoxData[ FormBox[ RowBox[{"exp", "(", "z", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " converges for all ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], ", it is legitimate to use term-by-term differentiation to find the \ derivative ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"exp", "'"}], RowBox[{"(", "z", ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49770043503125*^9, 3.497700464078125*^9}, { 3.49770052615625*^9, 3.497700563234375*^9}}], Cell[BoxData[{ RowBox[{"D", "[", RowBox[{ FractionBox[ SuperscriptBox["z", "n"], RowBox[{"n", "!"}]], ",", "z"}], "]"}], "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{ FractionBox[ SuperscriptBox["z", "0"], RowBox[{"0", "!"}]], ",", "z"}], "]"}]}], "Input", CellChangeTimes->{{3.497700679609375*^9, 3.4977007155625*^9}, { 3.497700763046875*^9, 3.4977007658125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], RowBox[{"D", "[", RowBox[{ FractionBox[ SuperscriptBox["z", "n"], RowBox[{"n", "!"}]], ",", "z"}], "]"}]}], "\[Equal]", RowBox[{"Exp", "[", "z", "]"}]}]], "Input", CellChangeTimes->{{3.497700728890625*^9, 3.4977007745*^9}}], Cell["Thus:", "Text", CellChangeTimes->{{3.497701734890625*^9, 3.497701737484375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"Exp", "[", "z", "]"}], ",", "z"}], "]"}], "\[Equal]", RowBox[{"Exp", "[", "z", "]"}]}]], "Input", CellChangeTimes->{{3.497701743234375*^9, 3.4977017513125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Basic law of exponents", "Subsection", CellChangeTimes->{{3.497700356140625*^9, 3.49770036190625*^9}}], Cell["The \"basic law\" of exponents is:", "Text", CellChangeTimes->{{3.49770036421875*^9, 3.49770039646875*^9}, { 3.4977017060625*^9, 3.49770171171875*^9}}], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"exp", "(", RowBox[{ SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]}], ")"}], "=", RowBox[{ RowBox[{"exp", "(", SubscriptBox["z", "1"], ")"}], " ", RowBox[{"exp", "(", SubscriptBox["z", "2"], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "EmphasisText", CellChangeTimes->{{3.49770036421875*^9, 3.49770039646875*^9}, { 3.4977017060625*^9, 3.49770171315625*^9}}], Cell[TextData[{ "To prove this, fix ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{ SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ":" }], "Text", CellChangeTimes->{{3.497700404078125*^9, 3.497700409296875*^9}, { 3.497701415625*^9, 3.497701462546875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"w", "=", RowBox[{ SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]}]}], ";"}]], "Input", CellChangeTimes->{{3.49770142540625*^9, 3.497701439015625*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"z", ",", "w"}], "]"}]], "Input", CellChangeTimes->{{3.497701558796875*^9, 3.497701585515625*^9}}], Cell[TextData[{ "Fix a complex number ", Cell[BoxData[ FormBox["w", TraditionalForm]], FormatType->"TraditionalForm"], " and define:" }], "Text", CellChangeTimes->{{3.497701445765625*^9, 3.497701448625*^9}, { 3.497701570421875*^9, 3.49770158228125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"g", "[", "z_", "]"}], ":=", RowBox[{ RowBox[{"Exp", "[", "z", "]"}], RowBox[{"Exp", "[", RowBox[{"w", "-", "z"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.497701467578125*^9, 3.49770147746875*^9}}], Cell["Then by the Product Rule and the Chain Rule:", "Text", CellChangeTimes->{{3.49770148753125*^9, 3.497701505203125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"g", "'"}], "[", "z", "]"}]], "Input", CellChangeTimes->{{3.497701490875*^9, 3.49770149221875*^9}}], Cell[TextData[{ "This means that ", Cell[BoxData[ FormBox["g", TraditionalForm]], FormatType->"TraditionalForm"], " is constant. But:" }], "Text", CellChangeTimes->{{3.497701510078125*^9, 3.497701518203125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"g", "[", "0", "]"}], "\[Equal]", RowBox[{"Exp", "[", "w", "]"}]}]], "Input", CellChangeTimes->{{3.49770152221875*^9, 3.497701523328125*^9}, { 3.497701600265625*^9, 3.497701604609375*^9}}], Cell[TextData[{ "Then \n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"Exp", "[", "z", "]"}], RowBox[{"Exp", "[", RowBox[{"w", "-", "z"}], "]"}]}], " ", "=", RowBox[{"Exp", "[", "w", "]"}], " "}], TraditionalForm]], FormatType->"TraditionalForm"], "\nfor all complex ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["w", TraditionalForm]], FormatType->"TraditionalForm"], ". Now take ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", SubscriptBox["z", "1"]}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{ SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49770161834375*^9, 3.497701687859375*^9}}, ParagraphSpacing->{0.5, 0}] }, Closed]], Cell[CellGroupData[{ Cell["Polar form of exp", "Subsection", CellChangeTimes->{{3.49770033403125*^9, 3.497700337859375*^9}}], Cell[TextData[{ "Consider first the Value of exp at an imaginary number. 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Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{"Re", " ", RowBox[{"(", RowBox[{"f", "(", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], ")"}]}]}], TraditionalForm]]], ", \t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{"Im", " ", RowBox[{"(", RowBox[{"f", "(", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], ")"}]}]}], TraditionalForm]]], ".\nSince the inputs to ", Cell[BoxData[ FormBox[ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " are ordered pairs of real numbers, each of the functions ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " may be regarded as real-valued functions whose domains are (subsets of) \ the complex plane. 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"[", RowBox[{"{", RowBox[{"realpart", ",", "imaginarypart"}], "}"}], "]"}]}], "Input", CellChangeTimes->{{3.4980516745625*^9, 3.498051900875*^9}, { 3.49805204353125*^9, 3.49805205528125*^9}, {3.498052096515625*^9, 3.49805212603125*^9}}], Cell[TextData[{ "Do you now \"see\" the periodicity of ", Cell[BoxData[ FormBox["exp", TraditionalForm]]], "\[LongDash]the property that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"exp", "(", RowBox[{"z", "+", RowBox[{"2", "\[Pi]", " ", "\[ImaginaryI]"}]}], ")"}], "=", RowBox[{"exp", "(", "z", ")"}]}], TraditionalForm]]], "\[LongDash]by looking at the preceding plot of the real and imaginary parts \ of ", Cell[BoxData[ FormBox["exp", TraditionalForm]]], "?" }], "Text", CellChangeTimes->{{3.49805207875*^9, 3.4980520828125*^9}, {3.498052343875*^9, 3.498052354625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". (a) What should three-dimensional plots of the modulus and principal \ argument of exp look like? Try to say before drawing them!\n(b) Now draw \ those plots.\n(c) Can you somehow \"see\" the periodicity of exp from these \ plots?" }], "Exercise", CellChangeTimes->{{3.49805240690625*^9, 3.498052479453125*^9}, { 3.498052597171875*^9, 3.498052665609375*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "The principal logarithm function ", Cell[BoxData[ FormBox["Log", TraditionalForm]]] }], "Section"], Cell[CellGroupData[{ Cell["Definition of the principal logarithm", "Subsection", CellChangeTimes->{{3.498057373078125*^9, 3.49805738253125*^9}, { 3.498057527703125*^9, 3.49805752803125*^9}, 3.498070449125*^9}], Cell[TextData[{ "The principal logarithm function ", Cell[BoxData[ FormBox["Log", TraditionalForm]]], " is a branch of the multivalued logarithm function and is defined for all \ complex ", Cell[BoxData[ FormBox[ RowBox[{"z", "\[NotEqual]", "0"}], TraditionalForm]]], " by:\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Log", "(", "z", ")"}], "=", RowBox[{"ln", "|", "z", "|", RowBox[{ RowBox[{"+", "\[ImaginaryI]"}], " ", RowBox[{"Arg", "(", "z", ")"}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.4980574333125*^9, 3.49805751775*^9}, {3.49806770175*^9, 3.498067706078125*^9}, 3.49807045590625*^9}, ParagraphSpacing->{0.5, 0}] }, Closed]], Cell[CellGroupData[{ Cell["Basic laws of logarithms", "Subsection", CellChangeTimes->{{3.49807061034375*^9, 3.498070615609375*^9}}], Cell[TextData[{ "The complex exponential function ", Cell[BoxData[ FormBox["exp", TraditionalForm]], FormatType->"TraditionalForm"], " is one-to-one only on its fundamental period strip ", Cell[BoxData[ FormBox[ RowBox[{"S", "=", RowBox[{"{", RowBox[{"z", ":", RowBox[{ RowBox[{"-", "\[Pi]"}], "<", RowBox[{"Im", RowBox[{"{", "z", "}"}]}], "\[LessEqual]", "\[Pi]"}]}], "}"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". Because of that, the \"usual\" laws of logarithms will not hold when some \ of the number involved are outside that fundamental period strip. For example:" }], "Text", CellChangeTimes->{{3.498070618703125*^9, 3.498070707296875*^9}, { 3.498070754515625*^9, 3.498070797796875*^9}, {3.49807093296875*^9, 3.4980709438125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Log", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[ImaginaryI]"}], ")"}], "\[ImaginaryI]"}], "]"}], "==", RowBox[{ RowBox[{"Log", "[", RowBox[{ RowBox[{"-", "1"}], "+", "\[ImaginaryI]"}], "]"}], "+", RowBox[{"Log", "[", "\[ImaginaryI]", "]"}]}]}]], "Input", CellChangeTimes->{{3.49807071365625*^9, 3.498070740484375*^9}, 3.498070843484375*^9, {3.49807095075*^9, 3.498070957640625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". (a) Calculate all three of the values of the principal logarithm there \ and explain why the logarithm of the product is not equal to the sum of the \ logarithms.\n(b) What is the actual relation between the values of the two \ sides? 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