(* 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[ 162338, 5290] NotebookOptionsPosition[ 136444, 4647] NotebookOutlinePosition[ 150818, 4976] CellTagsIndexPosition[ 150440, 4963] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Math 421 ", StyleBox["\[FilledSmallCircle]", FontSlant->"Plain"], " Fall 2010" }], "Subsubtitle", CellChangeTimes->{{3.491932055484375*^9, 3.4919320560625*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Cartesian and polar forms of complex numbers", "Subtitle", TextAlignment->Center, TextJustification->0], Cell["14 September 2010", "Subsubtitle", CellChangeTimes->{{3.490874635953125*^9, 3.49087464390625*^9}, 3.490907179140625*^9, 3.491167197609375*^9, {3.49193205175*^9, 3.49193205225*^9}, {3.492518926453125*^9, 3.492518929578125*^9}, { 3.493461250984375*^9, 3.4934612513125*^9}}, TextAlignment->Center], Cell["\<\ Based upon a notebook authored by David Park. New material and arrangement Copyright \[Copyright] 2004\[Dash]2010 by Murray \ Eisenberg. All rights reserved.\ \>", "SmallText", CellChangeTimes->{{3.49087462759375*^9, 3.490874627890625*^9}, 3.49087470490625*^9, {3.491932085203125*^9, 3.49193210259375*^9}, { 3.4919322084375*^9, 3.491932244375*^9}}, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "When you open this notebook, you should see a pop-up window asking whether \ you want to evaluate the Initialization Cells. You should select ", StyleBox["Yes", FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], "." }], "EmphasisText", CellChangeTimes->{ 3.490633836984375*^9, {3.49089844703125*^9, 3.490898447765625*^9}}], Cell[TextData[{ StyleBox["Explanation", FontSlant->"Italic"], ": In the ", ButtonBox["Initialization section", BaseStyle->"Hyperlink", ButtonData->"initialization"], " below, there's an Input cell consisting of the expression\n\t", StyleBox["<"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nand that cell has the property of being an \"initialization cell\". \ When you answered yes, that cell was evaluated automatically, and this in \ effect loaded the ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " application packages." }], "SmallText", CellChangeTimes->{{3.490633867953125*^9, 3.490633922609375*^9}, { 3.490650173484375*^9, 3.49065017828125*^9}, {3.490907105859375*^9, 3.490907106171875*^9}, {3.491166822984375*^9, 3.491166838875*^9}, { 3.491166882828125*^9, 3.491166882828125*^9}, {3.491919896484375*^9, 3.49191991115625*^9}, {3.491934281078125*^9, 3.49193430771875*^9}}, ParagraphSpacing->{0.7, 0}], Cell[CellGroupData[{ Cell["Prerequisites", "Section", CellChangeTimes->{{3.491159034765625*^9, 3.491159037671875*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematica", FontSlant->"Italic"]], "Subsection", CellChangeTimes->{{3.491159059828125*^9, 3.491159062328125*^9}}], Cell[TextData[{ "Some of the functions used in this notebook are built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions; others are provided by David Park's ", StyleBox["Presentations", FontSlant->"Italic"], " add-on. See the notebook ", StyleBox["AboutPresentations.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.49087489640625*^9, 3.490874908515625*^9}, { 3.49087660253125*^9, 3.49087661759375*^9}, {3.49087666071875*^9, 3.490876760109375*^9}, {3.490907300109375*^9, 3.490907301203125*^9}, { 3.491402168046875*^9, 3.491402170171875*^9}, {3.491402209734375*^9, 3.49140226371875*^9}, 3.4914022980625*^9, {3.49140250896875*^9, 3.491402516921875*^9}}], Cell[TextData[{ "You should already have ", StyleBox["Presentations", FontSlant->"Italic"], " available on the computer where you are using ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.49087489640625*^9, 3.490874908515625*^9}, { 3.49087660253125*^9, 3.49087661759375*^9}, {3.49087666071875*^9, 3.490876760109375*^9}, {3.490907300109375*^9, 3.490907301203125*^9}, { 3.491402168046875*^9, 3.491402170171875*^9}, {3.491402209734375*^9, 3.4914022918125*^9}}], Cell[TextData[{ "You should already know some of the basics about ", StyleBox["Mathematica", FontSlant->"Italic"], ": basic syntax; how to navigate around a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook; how to create an Input cell and type input into it; and how to \ evaluate an Input cell; how to use the Documentation Center for help; and how \ to open a palette." }], "Text", CellChangeTimes->{{3.4911590738125*^9, 3.491159285734375*^9}, { 3.491166982578125*^9, 3.491166982953125*^9}, {3.491402193890625*^9, 3.491402200390625*^9}, {3.4914037865*^9, 3.491403798703125*^9}}], Cell[TextData[{ "When you opened this notebook, you should have been prompted to let ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluate Initialization cells. That automatically loads ", StyleBox["Presentations", FontSlant->"Italic"], ". If you didn't allow that, you should load ", StyleBox["Presentations", FontSlant->"Italic"], " now: evaluate the initialization cell in the ", ButtonBox["Initialization section", BaseStyle->"Hyperlink", ButtonData->"initialization"], " below." }], "Text", CellChangeTimes->{{3.4911590738125*^9, 3.491159135453125*^9}, { 3.49115930428125*^9, 3.491159382515625*^9}, {3.4914019053125*^9, 3.491401908109375*^9}, {3.49140233146875*^9, 3.491402390734375*^9}, { 3.49140245890625*^9, 3.491402491609375*^9}, {3.49140259534375*^9, 3.49140264625*^9}}], Cell[TextData[{ "For information on how to get help about ", StyleBox["Presentations", FontSlant->"Italic"], ", see notebook ", StyleBox["AboutPresentations.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.49087489640625*^9, 3.490874908515625*^9}, { 3.49087660253125*^9, 3.49087661759375*^9}, {3.49087666071875*^9, 3.490876760109375*^9}, {3.490907300109375*^9, 3.490907301203125*^9}, { 3.4911589928125*^9, 3.49115901*^9}, {3.49140257375*^9, 3.4914025745625*^9}, 3.4914030055625*^9, {3.491934357390625*^9, 3.491934426421875*^9}}], Cell[TextData[{ "To read in the Documentation Center about the particular ", StyleBox["Presentations", FontSlant->"Italic"], " functions used in this notebook, open the \[OpenCurlyDoubleQuote]Complex \ Routines\[CloseCurlyDoubleQuote] group on the \ \[OpenCurlyDoubleQuote]ComplexGraphics\[CloseCurlyDoubleQuote] guide page. \ (Search for \[OpenCurlyDoubleQuote]ComplexGraphics\[CloseCurlyDoubleQuote].) " }], "Text", CellChangeTimes->{{3.4914028946875*^9, 3.491402922421875*^9}, { 3.491402970765625*^9, 3.49140302678125*^9}, {3.491403063953125*^9, 3.491403088296875*^9}, {3.491403616578125*^9, 3.491403653890625*^9}}], Cell[TextData[{ "The ", StyleBox["PresentationsPalette", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " is useful for accessing the documentation and for forming templates of ", StyleBox["Presentations", FontSlant->"Italic"], " functions. (For graphics here, you may also find useful the separate ", StyleBox["ComplexGraphics", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " palette. ) On ", StyleBox["PresentationsPalette", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], ":\n\t\[FilledSmallCircle] templates of functions to convert complex numbers \ between different forms are in ", StyleBox["Drawing \[RightPointer] Complex Graphics \[RightPointer] \ Coordinates", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], ";\n\t\[FilledSmallCircle] templates of functions to create complex graphics \ objects are in ", StyleBox["Drawing \[RightPointer] Complex Graphics \[RightPointer] \ Primitives", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], "; and\n\t\[FilledSmallCircle] templates of functions to create a drawing of \ complex graphics objects are in ", StyleBox["Drawing \[RightPointer] DrawingPaper \[RightPointer] Graphics", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " (the one you\[CloseCurlyQuote]ll want is ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ")." }], "Text", CellChangeTimes->{{3.491402670890625*^9, 3.49140273925*^9}, { 3.49140277784375*^9, 3.49140288615625*^9}, {3.491403685546875*^9, 3.491403769828125*^9}}, ParagraphSpacing->{0.5, 0}] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{{3.491159067078125*^9, 3.491159068140625*^9}}], Cell["\<\ You should already know the basics of the algebra of complex numbers and the \ representation of complex numbers by points in the plane.\ \>", "Text", CellChangeTimes->{{3.491166994109375*^9, 3.49116708359375*^9}, 3.491954728671875*^9}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section", ShowGroupOpener->True], Cell[TextData[{ "This notebook concerns the ", "Cartesian", " (", Cell[BoxData[ FormBox[ RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], TraditionalForm]]], ") and polar (", Cell[BoxData[ FormBox[ RowBox[{"r", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}], TraditionalForm]]], ") forms of complex numbers. It discusses some functions that allow you to \ convert between the two forms. In particular, it discusses how to perform \ such conversions when the input is a symbolic expression that represents \ complex numbers\[LongDash]a situation that you may find especially \ frustrating." }], "Text", CellChangeTimes->{{3.49087484890625*^9, 3.490874867625*^9}, { 3.490907219828125*^9, 3.490907220703125*^9}, {3.49193494396875*^9, 3.491934997359375*^9}, 3.492518903890625*^9}], Cell[TextData[{ "All the input shown uses the display form ", StyleBox["\[ImaginaryI]", FontFamily->"Courier"], " obtained by typing ", StyleBox["\[EscapeKey]", FontSize->16], StyleBox["ii", FontFamily->"Courier"], StyleBox["\[EscapeKey]", FontSize->16], ", but everything would work identically if you typed ", StyleBox["I", FontFamily->"Courier"], " instead." }], "Text", CellChangeTimes->{{3.491157816*^9, 3.4911578203125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section", ShowGroupOpener->True, CellChangeTimes->{3.4908745975625*^9}, CellTags->"initialization"], Cell[TextData[{ "Load David Park's ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " packages." }], "Text", CellChangeTimes->{{3.490874601625*^9, 3.490874618765625*^9}}], Cell[BoxData[ RowBox[{"<<", "Presentations`"}]], "Input", InitializationCell->True, CellChangeTimes->{{3.490874971453125*^9, 3.49087497475*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Cartesian form", "Section", ShowGroupOpener->True], Cell[CellGroupData[{ Cell[TextData[{ "Real and imaginary parts: ", StyleBox["Re", FontFamily->"Courier"], " and ", StyleBox["Im", FontFamily->"Courier"] }], "Subsection", ShowGroupOpener->True, CellChangeTimes->{{3.490907887546875*^9, 3.490907896953125*^9}}], Cell[TextData[{ "When a complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " is expressed as ", Cell[BoxData[ FormBox[ RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " with ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " real, that expression is called the ", "Cartesian", StyleBox[" form", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.49193407896875*^9, 3.49193416575*^9}, 3.492518904*^9}], Cell[TextData[{ "For a complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " given in ", "Cartesian", " form, its", StyleBox[" real part", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{"Re", "(", "x", ")"}], TraditionalForm]]], " is ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and its ", StyleBox["imaginary part", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{"Im", "(", "z", ")"}], TraditionalForm]]], " is ", Cell[BoxData[ FormBox["y", TraditionalForm]]], ". 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FontWeight->"Bold"], " Copy and modify the input for the preceding drawing in order to produce a \ similar drawing for the complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"-", "3"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{{3.491407791140625*^9, 3.4914078755*^9}, { 3.491408043984375*^9, 3.491408162109375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the real and imaginary parts of the sum ", Cell[BoxData[ FormBox[ RowBox[{"z", "+", "w"}], TraditionalForm]]], " of two complex numbers in terms of the real and imaginary parts of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["w", TraditionalForm]]], ".\n(b) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the real and imaginary parts of the multiple ", Cell[BoxData[ FormBox[ RowBox[{"\[Alpha]", " ", "z"}], TraditionalForm]]], " of a complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " by a real number ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]]], " in terms of the real and imaginary parts of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ".\n(c) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the real and imaginary parts of the multiple ", Cell[BoxData[ FormBox[ RowBox[{"\[ImaginaryI]", " ", "z"}], TraditionalForm]]], " of a complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " in terms of the real and imaginary parts of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ".\n(d) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the real and imaginary parts of the product ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", "w"}], TraditionalForm]]], " of two complex numbers in terms of the real and imaginary parts of the \ numbers ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["w", TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{{3.491407791140625*^9, 3.4914078755*^9}, { 3.491408043984375*^9, 3.491408162109375*^9}, {3.491415853890625*^9, 3.491416164515625*^9}}, ParagraphSpacing->{0.5, 0.}, CellTags->"exercReImProperties"], Cell[TextData[{ "An object such as ", StyleBox["2+3\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is regarded by ", StyleBox["Mathematica", FontSlant->"Italic"], " as being numeric:" }], "Text", CellChangeTimes->{{3.49087506209375*^9, 3.4908750638125*^9}}], Cell[BoxData[ RowBox[{"NumericQ", "[", "z", "]"}]], "Input"], Cell[TextData[{ "And such an object is actually stored ", StyleBox["not", FontSlant->"Italic"], " as a sum of its real part and \[ImaginaryI] times its imaginary part, but \ rather as an object of type ", StyleBox["Complex", FontFamily->"Courier"], ":" }], "Text"], Cell[BoxData[{ RowBox[{"Head", "[", "z", "]"}], "\n", RowBox[{"FullForm", "[", "z", "]"}]}], "Input"], Cell[TextData[{ "Difficulty does arise when you have only a ", StyleBox["symbolic", FontSlant->"Italic"], " expression representing complex numbers. For example:" }], "Text", CellChangeTimes->{3.490907401921875*^9}], Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"x", ",", "y"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "z", "]"}], ",", RowBox[{"Im", "[", "z", "]"}]}], "}"}]}], "Input", CellChangeTimes->{{3.49087508346875*^9, 3.4908751128125*^9}}], Cell[BoxData[ RowBox[{"NumericQ", "[", "z", "]"}]], "Input", CellChangeTimes->{{3.490876090953125*^9, 3.49087609165625*^9}}], Cell[TextData[{ "The reason is that, ", StyleBox["by default, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" regards all symbolic variables representing numbers to be complex", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "!" }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489419769390625*^9, 3.489419937828125*^9}, { 3.489420016078125*^9, 3.489420017734375*^9}, {3.48942006003125*^9, 3.489420061125*^9}, {3.490280766890625*^9, 3.49028080634375*^9}}], Cell[TextData[{ "You have to explicitly tell ", StyleBox["Mathematica", FontSlant->"Italic"], " when you want all such variables in an expression to be regarded, instead, \ as real. Do so by using the built-in function ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". See the ", ButtonBox["next section", BaseStyle->"Hyperlink", ButtonData->"FunctionComplexExpand"], "." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489419769390625*^9, 3.489419937828125*^9}, { 3.489420016078125*^9, 3.489420091171875*^9}, {3.490283492734375*^9, 3.49028349321875*^9}, {3.490875457546875*^9, 3.4908754719375*^9}, { 3.49192004084375*^9, 3.491920065359375*^9}, {3.49192010628125*^9, 3.491920106296875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Designating variables as real: ", StyleBox["ComplexExpand", FontFamily->"Courier"] }], "Subsection", ShowGroupOpener->True, CellChangeTimes->{{3.490876871984375*^9, 3.490876888796875*^9}, 3.49090790428125*^9}, CellTags->"FunctionComplexExpand"], Cell[TextData[{ "To treat all symbolic variables in an expression for a complex number to be \ real, use ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "." }], "EmphasisText", CellChangeTimes->{{3.491920395546875*^9, 3.49192043146875*^9}}], Cell[BoxData[ RowBox[{" ", RowBox[{"?", "ComplexExpand"}]}]], "Input"], Cell["Look again at the symbolic example:", "Text"], Cell[BoxData[ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}]], "Input", CellChangeTimes->{{3.490875144125*^9, 3.49087514528125*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", "z", "]"}]], "Input"], Cell["\<\ You can take its real and imaginary parts like this\[Ellipsis]\ \>", "Text", CellChangeTimes->{{3.490875159109375*^9, 3.490875163875*^9}, { 3.491404074046875*^9, 3.491404079109375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", "z", "]"}], "]"}], ",", RowBox[{"ComplexExpand", "[", RowBox[{"Im", "[", "z", "]"}], "]"}]}], "}"}]], "Input", CellChangeTimes->{3.4908751925625*^9}], Cell["\[Ellipsis]or, more simply, like this:", "Text", CellChangeTimes->{{3.490875168953125*^9, 3.490875174125*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "z", "]"}], ",", RowBox[{"Im", "[", "z", "]"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.490875175875*^9, 3.490875189*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " You saw that\n\t", StyleBox["{ComplexExpand[Re[z]],ComplexExpand[Im[z]]}\n", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "gives the real and imaginary parts of a symbolic expression. What happens \ if you use, instead, the following?\n\t", StyleBox["{Re[ComplexExpand[z]],Im[ComplexExpand[z]]}\n", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Why?", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"] }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.4908803198125*^9, 3.49088045078125*^9}, {3.490880480921875*^9, 3.49088057415625*^9}, { 3.4909074508125*^9, 3.49090745478125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[CellGroupData[{ Cell[TextData[{ "Assumed complex real variables and target functions in ", StyleBox["ComplexExpand", FontFamily->"Courier"], " (", StyleBox["advanced", FontSlant->"Italic"], ")" }], "Subsubsection", ShowGroupOpener->True, CellChangeTimes->{{3.490875494578125*^9, 3.490875515625*^9}}, CellTags->"AdvancedComplexExpand"], Cell[TextData[{ "When used with one argument, ", StyleBox["ComplexExpand", FontFamily->"Courier"], " assumes that all variables are real. You may explicitly specify that some \ are to be regarded as complex by including them in a list that forms a second \ argument. For example:" }], "Text", CellChangeTimes->{3.490875227765625*^9, 3.49087612759375*^9}], Cell[BoxData[ RowBox[{"w", "=", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "\[Zeta]"}], ")"}], "2"], " ", SuperscriptBox["t", "2"]}], "+", RowBox[{"\[ImaginaryI]", RowBox[{"(", RowBox[{"2", "-", "s"}], ")"}], RowBox[{"(", RowBox[{"\[Xi]", "+", "1"}], ")"}]}]}]}]], "Input", CellChangeTimes->{{3.490875534015625*^9, 3.49087555453125*^9}}], Cell[TextData[{ "(The symbols \[Zeta] and \[Xi], Greek letters \[OpenCurlyDoubleQuote]zeta\ \[CloseCurlyDoubleQuote] and \ \[OpenCurlyDoubleQuote]xi\[CloseCurlyDoubleQuote], may be typed as ", StyleBox["\[EscapeKey]z\[EscapeKey]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["\[EscapeKey]x\[EscapeKey]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", respectively.)" }], "Text", CellChangeTimes->{{3.490876138046875*^9, 3.49087620871875*^9}, 3.490876239171875*^9}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"w", ",", RowBox[{"{", RowBox[{"\[Zeta]", ",", "\[Xi]"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.490875561375*^9, 3.490875564671875*^9}}], Cell[TextData[{ "Usually, ", StyleBox["ComplexExpand", FontFamily->"Courier"], " will try to express its result so as to involve \"target functions\" ", StyleBox["Re", FontFamily->"Courier"], " and ", StyleBox["Im", FontFamily->"Courier"], ". But you can force it to use other functions by means of the option ", StyleBox["TargetFunctions", FontFamily->"Courier"], ". For example:" }], "Text", CellChangeTimes->{{3.490875582125*^9, 3.4908755900625*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"1", "/", "\[Zeta]"}], ",", RowBox[{"{", "\[Zeta]", "}"}]}], "]"}]], "Input", CellChangeTimes->{3.490875594703125*^9}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"1", "/", "\[Zeta]"}], ",", RowBox[{"{", "\[Zeta]", "}"}], ",", RowBox[{"TargetFunctions", "\[Rule]", RowBox[{"{", RowBox[{"Abs", ",", "Conjugate"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.490875603625*^9, 3.4908756169375*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"\[Zeta]", " ", RowBox[{"Conjugate", "[", "\[Zeta]", "]"}]}], ",", RowBox[{"{", "\[Zeta]", "}"}]}], "]"}]], "Input", CellChangeTimes->{3.4908756259375*^9}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"\[Zeta]", " ", RowBox[{"Conjugate", "[", "\[Zeta]", "]"}]}], ",", RowBox[{"{", "\[Zeta]", "}"}], ",", RowBox[{"TargetFunctions", "\[Rule]", "Abs"}]}], "]"}]], "Input", CellChangeTimes->{{3.49087563671875*^9, 3.490875646828125*^9}}], Cell[TextData[{ "For more about such advanced uses of ", StyleBox["ComplexExpand", FontFamily->"Courier"], ", see", ButtonBox[" below", BaseStyle->"Hyperlink", ButtonData:>"MoreAdvancedComplexExpand"], "." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Converting from ", StyleBox["x+\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "\[ThinSpace]", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " to ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], FontFamily->"Courier"], TraditionalForm]]], ": ", StyleBox["ToCoordinates", FontFamily->"Courier"] }], "Subsection", ShowGroupOpener->True, CellChangeTimes->{{3.4908756745625*^9, 3.49087568228125*^9}, { 3.49087682678125*^9, 3.4908768526875*^9}, {3.490876901828125*^9, 3.490876903796875*^9}, {3.49088076953125*^9, 3.490880917625*^9}, { 3.49192025421875*^9, 3.49192025609375*^9}, 3.491920345140625*^9, { 3.493461336578125*^9, 3.49346134615625*^9}}], Cell[TextData[{ "The ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ToCoordinates", FontFamily->"Courier"], " provides a shortcut for ", StyleBox["{Re[z],Im[z]}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.490876913*^9, 3.490876922203125*^9}, { 3.49087704728125*^9, 3.490877101796875*^9}}], Cell[BoxData[ RowBox[{"?", "ToCoordinates"}]], "Input"], Cell["For example:", "Text"], Cell[BoxData[ RowBox[{"ToCoordinates", "[", RowBox[{"2", "+", RowBox[{"3", "\[ImaginaryI]"}]}], "]"}]], "Input", CellChangeTimes->{{3.4908769355625*^9, 3.490876936578125*^9}}], Cell["Recall again the symbolic example:", "Text"], Cell[BoxData[ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}]], "Input", CellChangeTimes->{{3.49087695534375*^9, 3.490876975125*^9}}], Cell["Try this:", "Text"], Cell[BoxData[ RowBox[{"ToCoordinates", "[", "z", "]"}]], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " still does not know that you intend ", StyleBox["x", FontFamily->"Courier"], " and ", StyleBox["y", FontFamily->"Courier"], " to be real. And again ", StyleBox["ComplexExpand", FontFamily->"Courier"], " comes to the rescue:" }], "Text", CellChangeTimes->{{3.4908769836875*^9, 3.49087700678125*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"ToCoordinates", "[", "z", "]"}], "]"}]], "Input"], Cell[TextData[{ "To convert a symbolic complex quantity ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to the ordered pair consisting of its real and imaginary parts, use:\n\t", StyleBox["ComplexExpand[ToCoordinates[z]]", FontFamily->"Courier"] }], "EmphasisText", CellChangeTimes->{{3.49087714284375*^9, 3.490877188046875*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "Here's the same thing using function composition ", StyleBox["@", FontFamily->"Courier"], " notation (also called \[OpenCurlyDoubleQuote]prefix notation\ \[CloseCurlyDoubleQuote]) to eliminate nested brackets:" }], "Text", CellChangeTimes->{{3.49140880509375*^9, 3.4914088159375*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "@", RowBox[{"ToCoordinates", "[", "z", "]"}]}]], "Input", CellChangeTimes->{{3.49087724978125*^9, 3.490877263828125*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " What happens if you use the opposite order: ", StyleBox["ToCoordinates[ComplexExpand[z]]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "? Why doesn\[CloseCurlyQuote]t that yield the same result as ", StyleBox["ComplexExpand[ToCoordinates[z]]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490877288703125*^9, 3.49087748725*^9}, {3.490877875234375*^9, 3.49087791125*^9}, 3.490880664921875*^9}], Cell[TextData[{ "Sometimes you want to bring in ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as an \[OpenCurlyDoubleQuote]afterthought\[CloseCurlyDoubleQuote] to the \ main expression. Then you may use the \[OpenCurlyDoubleQuote]postfix\ \[CloseCurlyDoubleQuote] form of input, with ", StyleBox["//", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490283603515625*^9, 3.490283658703125*^9}, { 3.49028376921875*^9, 3.49028377653125*^9}, {3.490283913453125*^9, 3.490283925515625*^9}, {3.49087793615625*^9, 3.490877941859375*^9}, { 3.490877988375*^9, 3.49087798934375*^9}, {3.49192277825*^9, 3.491922780734375*^9}}, CellID->321563456], Cell[BoxData[{ RowBox[{"z", "//", "ComplexExpand"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"ToCoordinates", "[", "z", "]"}], "//", "ComplexExpand"}]}], "Input", CellChangeTimes->{ 3.48969559478125*^9, {3.490283660921875*^9, 3.49028366771875*^9}, { 3.49028393571875*^9, 3.490283936046875*^9}, {3.49087794740625*^9, 3.490877976625*^9}}, CellID->107301876], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Repeat Exercise ", CounterBox["Exercise", "exercReImProperties"], " but now using ", StyleBox["ToCoordinates", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to simplify the work." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49140891653125*^9, 3.49140896528125*^9}, {3.491409017828125*^9, 3.4914090435625*^9}, { 3.4914161915*^9, 3.491416193078125*^9}, {3.49151302490625*^9, 3.491513029828125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Converting from From", StyleBox[" ", FontFamily->"Times"], StyleBox["{x,y}", FontFamily->"Courier"], " to ", StyleBox["x+\[ImaginaryI]\[ThinSpace]y", FontFamily->"Courier"], ": ", StyleBox["ToComplex", FontFamily->"Courier"] }], "Subsection", ShowGroupOpener->True, CellChangeTimes->{{3.490880887921875*^9, 3.4908808918125*^9}, { 3.49088094990625*^9, 3.490880967390625*^9}, {3.4934613605*^9, 3.49346136509375*^9}}], Cell["\<\ To go backwards from the list of its real and imaginary parts to a complex \ number presents no difficulty, even for symbolic expressions.\ \>", "Text", CellChangeTimes->{{3.49088099159375*^9, 3.49088101753125*^9}}], Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"xy", ",", "s", ",", "t"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"xy", "=", RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"2", " ", "s"}], "+", SuperscriptBox["s", "2"]}], ",", RowBox[{"4", "-", RowBox[{"4", " ", "s", " ", "t"}], "+", RowBox[{ SuperscriptBox["s", "2"], " ", SuperscriptBox["t", "2"]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"First", "[", "xy", "]"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Last", "[", "xy", "]"}]}]}]}], "Input", CellChangeTimes->{{3.490881021453125*^9, 3.490881036984375*^9}}], Cell[TextData[{ " The ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ToComplex", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " makes that easier to do." }], "Text", CellChangeTimes->{{3.490881045125*^9, 3.490881051078125*^9}}], Cell[BoxData[ RowBox[{"?", "ToComplex"}]], "Input"], Cell[BoxData[ RowBox[{"ToComplex", "[", RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"2", " ", "s"}], "+", SuperscriptBox["s", "2"]}], ",", RowBox[{"4", "-", RowBox[{"4", " ", "s", " ", "t"}], "+", RowBox[{ SuperscriptBox["s", "2"], " ", SuperscriptBox["t", "2"]}]}]}], "}"}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Are the functions ", StyleBox["ToComplex", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["ToCoordinates", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " inverses of one another\[LongDash]that is, does each undo the effect of \ other? In arriving at an answer, consider first specific numerical complex \ numbers such as ", StyleBox["z=2+3\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and then symbolic complex values, such as ", Cell[BoxData[ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}]], CellChangeTimes->{3.4908812085*^9}], "?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490881094203125*^9, 3.490881238078125*^9}, {3.49088136890625*^9, 3.490881514234375*^9}, { 3.490881614390625*^9, 3.4908816650625*^9}, {3.49088172240625*^9, 3.490881731140625*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Conjugate", "Section", ShowGroupOpener->True], Cell[TextData[{ "The ", StyleBox["conjugate", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of a complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " with ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], " real is the complex number ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"z", "\[Conjugate]"}], "=", RowBox[{"x", "-", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49140919375*^9, 3.49140923478125*^9}}], Cell[TextData[{ "The built-in function ", StyleBox["Conjugate", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " finds\[LongDash]you guessed it!\[LongDash]the conjugate of a complex \ number." }], "Text"], Cell[BoxData[ RowBox[{"?", "Conjugate"}]], "Input"], Cell[BoxData[ RowBox[{"Conjugate", "[", RowBox[{"2", "+", RowBox[{"3", " ", "I"}]}], "]"}]], "Input", CellChangeTimes->{{3.490881792421875*^9, 3.49088179375*^9}}], Cell["\<\ As usual, there's difficulty when you don't have a specific complex number, \ but rather a symbolic expression representing complex numbers:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.490881811359375*^9, 3.490881826640625*^9}}], Cell[BoxData[ RowBox[{"Conjugate", "[", "z", "]"}]], "Input"], Cell[TextData[{ "The simplest way to handle this is again with ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{3.490881841*^9, {3.490907994046875*^9, 3.490907994625*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"Conjugate", "[", "z", "]"}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find the conjugate of ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491409118625*^9, 3.49140915021875*^9}, 3.491416245046875*^9, {3.491416782515625*^9, 3.491416788515625*^9}, {3.491417131359375*^9, 3.4914171406875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " For an arbitrary complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " with ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], " real, find the real and imaginary parts of the conjugate of ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491409118625*^9, 3.49140915021875*^9}, 3.491416245046875*^9, {3.491416782515625*^9, 3.491416788515625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Let ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{"c", "+", RowBox[{"d", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], ".\n(a) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the conjugate of the sum ", Cell[BoxData[ FormBox[ RowBox[{"z", "+", "w"}], TraditionalForm]]], " in terms of the conjugates of ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], ".\n(b) Repeat (a) but for the product ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", "w"}], TraditionalForm]]], ".\n(c) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the conjugate of ", Cell[BoxData[ FormBox[ RowBox[{"-", "z"}], TraditionalForm]]], " in terms of the conjugate of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ".\n(c) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to express the conjugate of ", Cell[BoxData[ FormBox[ RowBox[{"\[ImaginaryI]", " ", "z"}], TraditionalForm]]], " in terms of the conjugate of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491409118625*^9, 3.49140915021875*^9}, 3.491416245046875*^9, {3.491416803125*^9, 3.491416882875*^9}, {3.491416925578125*^9, 3.491416951671875*^9}, { 3.49141718971875*^9, 3.491417195640625*^9}, {3.4914172519175777`*^9, 3.4914172729175777`*^9}}, ParagraphSpacing->{0.5, 0.}, CellTags->"exercConjProperties"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) What is the geometric relationship between a complex number and its \ conjugate?\n(b) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a drawing that illustrates this relationship." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491409118625*^9, 3.49140915021875*^9}, 3.491416245046875*^9, {3.4914162964375*^9, 3.491416364703125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[CellGroupData[{ Cell["\<\ Traditional notation for conjugate\ \>", "Subsection", CellChangeTimes->{{3.490908232859375*^9, 3.49090823884375*^9}}], Cell[TextData[{ "Two standard mathematical notations are in use for the conjugate of a \ complex number ", Cell[BoxData[ FormBox[Cell["z"], TraditionalForm]]], ", namely, with an overbar ( ", Cell[BoxData[ FormBox[ OverscriptBox["z", "_"], TraditionalForm]]], " ) and with a superscript star ( ", Cell[BoxData[ FormBox[ RowBox[{"z", "\[Conjugate]"}], TraditionalForm]]], " ). Note that the superscript star in the latter is not the same as an \ asterisk (", StyleBox["*)", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ": the former has 6 points whereas the latter has only 5. (To see the \ difference, you may need to magnify this notebook.)" }], "Text", CellChangeTimes->{{3.490882860484375*^9, 3.490882892265625*^9}, { 3.490883038421875*^9, 3.49088306203125*^9}, {3.490883094140625*^9, 3.490883349984375*^9}, {3.490907935875*^9, 3.490907958796875*^9}}], Cell[TextData[{ "To type the superscript star notation conjugate, type \[EscapeKey]", StyleBox["conj", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[EscapeKey] after the expression whose conjugate you want. But do ", StyleBox["not", FontSlant->"Italic"], " use \[ControlKey]", StyleBox["^", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to get a superscript: ", StyleBox["Mathematica", FontSlant->"Italic"], " will position the star automatically. ", StyleBox["Try it", Background->RGBColor[0.941, 0.973, 1.]], ": position the cursor at the end of the ", StyleBox["second line", FontSlant->"Italic"], " of following cell, immediately after the z; then type \[EscapeKey]conj\ \[EscapeKey] and evaluate the cell." }], "Text", CellChangeTimes->{{3.490908290953125*^9, 3.490908329421875*^9}, { 3.490908375625*^9, 3.490908469578125*^9}, {3.490908558140625*^9, 3.490908675625*^9}, {3.491416508625*^9, 3.4914165216875*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"z", "=", RowBox[{"2", "+", RowBox[{"3", "I"}]}]}], ";"}], "\[IndentingNewLine]", "z"}], "Input", CellChangeTimes->{{3.49090868071875*^9, 3.49090869203125*^9}, { 3.491416490640625*^9, 3.491416533109375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) In a new input cell, type ", StyleBox["2+3\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[EscapeKey]", StyleBox["conj", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[EscapeKey] and evaluate the cell.\n(b) Repeat (a) but with ", StyleBox["3\[ImaginaryI]+2", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " instead of ", StyleBox["2+3\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Explain why the result is different.\n(c) Without changing ", StyleBox["3\[ImaginaryI]+2", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", what else should you type so as to force the star to take the conjugate \ of the entire complex number?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490908725109375*^9, 3.49090886284375*^9}, {3.490908912421875*^9, 3.4909089276875*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " When you use the superscript star (\[EscapeKey]", StyleBox["conj", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[EscapeKey]) form for ", StyleBox["Conjugate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", how do you get ", StyleBox["Mathematica", FontSlant->"Italic"], " to find symbolically the conjugate of a complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " where ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " are real?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490908725109375*^9, 3.49090886284375*^9}, {3.490908912421875*^9, 3.4909089276875*^9}, { 3.4909090236875*^9, 3.49090909240625*^9}, {3.491416570609375*^9, 3.4914167439375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Repeat Exercise ", CounterBox["Exercise", "exercConjProperties"], ", but now using the superscript star notation." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490908725109375*^9, 3.49090886284375*^9}, {3.490908912421875*^9, 3.4909089276875*^9}, { 3.4909090236875*^9, 3.49090909240625*^9}, {3.491416570609375*^9, 3.4914167439375*^9}, {3.491513578328125*^9, 3.4915136174375*^9}}], Cell[TextData[{ "If you want to type an overbar to indicate conjugate, after the quantity \ whose conjugate you want type: \[ControlKey]", StyleBox["&_", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " The last symbol is the underscore character, typed as usual as \ \[ShiftKey]", StyleBox["-", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " .\nFor example, ", Cell[BoxData[ FormBox[ OverscriptBox[ RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", "\[ImaginaryI]"}]}], ")"}], "_"], TraditionalForm]]], " is typed as\n\t", StyleBox["(2+3", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[EscapeKey]", StyleBox["ii", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[EscapeKey]", StyleBox[")", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[ControlKey]", StyleBox["&_", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.490909216328125*^9, 3.4909093965625*^9}, { 3.490909455875*^9, 3.490909609734375*^9}, {3.4909096759375*^9, 3.49090971440625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "and ", Cell[BoxData[ FormBox[ OverscriptBox["z", "_"], TraditionalForm]], FormatType->"TraditionalForm"], " is typed as: ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[ControlKey]", StyleBox["&_", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" .", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.490909216328125*^9, 3.4909093965625*^9}, { 3.490909455875*^9, 3.490909605*^9}, {3.4909096385625*^9, 3.490909673046875*^9}, {3.490909717609375*^9, 3.490909737828125*^9}}], Cell[TextData[{ "However, ", StyleBox["Mathematica", FontSlant->"Italic"], " will ", StyleBox["not", FontSlant->"Italic"], " interpret overbar notation to mean complex conjugate. For example, \ evaluate the following:" }], "Text", CellChangeTimes->{{3.490909216328125*^9, 3.4909093965625*^9}, { 3.490909455875*^9, 3.490909605*^9}, {3.4909096385625*^9, 3.490909673046875*^9}, {3.490909717609375*^9, 3.490909767265625*^9}}], Cell[BoxData[ OverscriptBox[ RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", "\[ImaginaryI]"}]}], ")"}], "_"]], "Input", CellChangeTimes->{{3.49090977278125*^9, 3.490909779109375*^9}}], Cell[TextData[{ "It is possible to force ", StyleBox["Mathematica", FontSlant->"Italic"], " to interpret the overbar notation, too, as meaning conjugate. But trying \ to do that may be more trouble than it\[CloseCurlyQuote]s worth, as the \ overbar tends to typeset too low." }], "SmallText", CellChangeTimes->{{3.490909783578125*^9, 3.490909913578125*^9}, { 3.4914173271363277`*^9, 3.4914173986832027`*^9}, {3.4914175595425777`*^9, 3.4914175681207027`*^9}, 3.491922901671875*^9}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Conjugate by using a replacement rule (", StyleBox["advanced", FontSlant->"Italic"], ")" }], "Subsection", CellChangeTimes->{{3.490908039640625*^9, 3.49090804903125*^9}, { 3.4909091575625*^9, 3.490909159515625*^9}}], Cell[TextData[{ "Another way to obtain the conjugate in the form you probably want is to use \ a ", StyleBox["rule", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ":" }], "Text", CellChangeTimes->{{3.490881966109375*^9, 3.490881968625*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"z", "/.", RowBox[{ RowBox[{"u_", "+", RowBox[{"\[ImaginaryI]", " ", "v_"}]}], "\[Rule]", RowBox[{"u", "-", RowBox[{"\[ImaginaryI]", " ", "v"}]}]}]}]}], "Input", CellChangeTimes->{{3.49088198565625*^9, 3.490882007296875*^9}, { 3.491922971*^9, 3.491922971640625*^9}}], Cell[TextData[{ "A tempting way to use a rule to form the conjugate is to replace ", StyleBox["\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in the complex expression with ", StyleBox["-\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Try it:" }], "Text", CellChangeTimes->{{3.49088202971875*^9, 3.49088212375*^9}, { 3.4908825488125*^9, 3.490882575984375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"2", "+", RowBox[{"3", "\[ImaginaryI]"}]}], "/.", RowBox[{"\[ImaginaryI]", "\[Rule]", RowBox[{"-", "\[ImaginaryI]", " ", RowBox[{"(*", " ", RowBox[{"does", " ", "not", " ", RowBox[{"work", "!"}]}], " ", "*)"}]}]}]}]], "Input", CellChangeTimes->{{3.490882126765625*^9, 3.4908821370625*^9}, { 3.490882698984375*^9, 3.49088271321875*^9}}], Cell[TextData[{ StyleBox["\[WarningSign]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0]], " ", StyleBox["Caution:", FontSlant->"Italic"], " What\[LongDash]no change? Not even with a specific, numeric complex \ number! Why not? While ", StyleBox["Mathematica ", FontSlant->"Italic"], "displays a complex number as a sum ", StyleBox["a+b\[ThinSpace]\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", that is ", StyleBox["not", FontSlant->"Italic"], " the way it actually represents the complex number internally. Here\ \[CloseCurlyQuote]s the way ", StyleBox["Mathematica", FontSlant->"Italic"], " represents the complex number:" }], "Text", CellChangeTimes->{{3.49088214521875*^9, 3.49088246878125*^9}, 3.490882592484375*^9, {3.49192298896875*^9, 3.49192306640625*^9}}], Cell[BoxData[ RowBox[{"FullForm", "[", RowBox[{"2", "+", RowBox[{"3", "\[ImaginaryI]"}]}], "]"}]], "Input", CellChangeTimes->{{3.490882207296875*^9, 3.490882214765625*^9}, { 3.490882603640625*^9, 3.490882604390625*^9}}], Cell[TextData[{ "As you can plainly see, there is no ", StyleBox["\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " (or ", StyleBox["I", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ") in that representation. And so there is no ", StyleBox["\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to replace when you try the rule ", StyleBox["\[ImaginaryI]\[Rule]-\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". This is a \[OpenCurlyDoubleQuote]gotcha\[CloseCurlyDoubleQuote] that \ sometimes confounds ", StyleBox["Mathematica", FontSlant->"Italic"], " novices." }], "Text", CellChangeTimes->{{3.4908824730625*^9, 3.490882538203125*^9}, { 3.49088261471875*^9, 3.49088261871875*^9}, {3.490882650640625*^9, 3.49088266440625*^9}, {3.49088272846875*^9, 3.4908827383125*^9}, { 3.49090917528125*^9, 3.490909189078125*^9}, {3.4914249640113277`*^9, 3.4914249669332027`*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Polar form", "Section", ShowGroupOpener->True], Cell[CellGroupData[{ Cell[TextData[{ "Modulus and principal argument: ", StyleBox["Abs", FontFamily->"Courier"], " and ", StyleBox["Arg", FontFamily->"Courier"] }], "Subsection", ShowGroupOpener->True, CellChangeTimes->{{3.49091001309375*^9, 3.490910037921875*^9}, { 3.49193450225*^9, 3.491934503328125*^9}}], Cell[TextData[{ "The ", StyleBox["modulus", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{"|", "z", "|"}], TraditionalForm]], FormatType->"TraditionalForm"], " of a complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", where ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], " are real, is ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}]], TraditionalForm]], FormatType->"TraditionalForm"], ". Geometrically, ", Cell[BoxData[ FormBox[ RowBox[{"|", "z", "|"}], TraditionalForm]], FormatType->"TraditionalForm"], " is the distance between the point ", Cell[BoxData[ FormBox[ RowBox[{"z", " "}], TraditionalForm]], FormatType->"TraditionalForm"], " and the origin of the complex plane. " }], "Text", CellChangeTimes->{{3.4914250157457027`*^9, 3.4914251360425777`*^9}, { 3.4914251840113277`*^9, 3.4914251865269527`*^9}, {3.4914259261050777`*^9, 3.4914259304332027`*^9}}], Cell[TextData[{ "An ", StyleBox["argument", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of a complex number is the radian measure ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]]], " of an angle between the positive real axis and the line segment connecting \ the origin to the point ", Cell[BoxData[ FormBox["z", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4914251411207027`*^9, 3.4914253063082027`*^9}, { 3.4914259380425777`*^9, 3.4914259438394527`*^9}, {3.491923177203125*^9, 3.49192319128125*^9}, 3.491934517109375*^9}], Cell[TextData[{ "Note that the terminology \[OpenCurlyDoubleQuote]", StyleBox["an", FontSlant->"Italic"], " argument\[CloseCurlyDoubleQuote] above. That\[CloseCurlyQuote]s because if \ ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]]], " is an argument of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ", then so are all the numbers ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "+", RowBox[{"2", "k", " ", "\[Pi]"}]}], TraditionalForm]]], ", for ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"k", "=", RowBox[{"\[PlusMinus]", "1"}]}], ",", RowBox[{"\[PlusMinus]", "2"}], ",", " ", RowBox[{"\[PlusMinus]", "3"}], ",", " ", "\[Ellipsis]"}], TraditionalForm]]], ". The infinite set of all arguments of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " is denoted by ", Cell[BoxData[ FormBox[ RowBox[{"arg", "(", "z"}], TraditionalForm]]], "). " }], "Text", CellChangeTimes->{{3.4914254653863277`*^9, 3.4914256032457027`*^9}, { 3.4914256580582027`*^9, 3.4914256676988277`*^9}, {3.4914259723707027`*^9, 3.4914260353550777`*^9}, {3.4914260982769527`*^9, 3.4914261006207027`*^9}, { 3.491923204453125*^9, 3.491923227421875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Let ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}]}], TraditionalForm]]], " with ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " real. If ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"|", "z", "|"}]}], TraditionalForm]]], ", the modulus of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ", and if \[Theta] is an argument of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ", then of course the pair ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"r", ",", "\[Theta]"}], ")"}], TraditionalForm]]], " constitute polar coordinates of the point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], ". 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Check your answer with ", StyleBox["Mathematica", FontSlant->"Italic"], ", both for a specifc numeric ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], " and for an arbitrary symbolic ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", "y\[ImaginaryI]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ".\n(b) If ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], " is complex, express ", Cell[BoxData[ FormBox[ RowBox[{"Arg", "(", RowBox[{"\[ImaginaryI]", " ", "z"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " in terms of ", Cell[BoxData[ FormBox[ RowBox[{"Arg", "(", "z", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ". 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For example:" }], "Text", CellChangeTimes->{{3.490910377359375*^9, 3.490910410390625*^9}, { 3.490910840734375*^9, 3.49091091884375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Arg", "[", RowBox[{ RowBox[{"-", "1"}], "+", "\[ImaginaryI]"}], "]"}], ",", RowBox[{"ArcTan", "[", FractionBox["1", RowBox[{"-", "1"}]], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.49091059453125*^9, 3.490910598234375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Arg", "[", RowBox[{ RowBox[{"-", "1"}], "-", "\[ImaginaryI]"}], "]"}], ",", RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"-", "1"}], RowBox[{"-", "1"}]], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.490910620359375*^9, 3.49091062434375*^9}}], Cell[TextData[{ StyleBox["\[WarningSign]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0]], StyleBox[" Caution:", FontSlant->"Italic"], " It is very dangerous to use the ", StyleBox["Mathematica", FontSlant->"Italic"], " expression ", StyleBox["ArcTan[y/x]", FontFamily->"Courier"], " to find ", StyleBox["Arg[x+\[ImaginaryI]\[ThinSpace]y]", FontFamily->"Courier"], StyleBox[", even when ", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["x", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" and ", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["y", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" have specific numerical values.", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.49091064303125*^9, 3.49091069409375*^9}, { 3.49192352190625*^9, 3.4919235276875*^9}}], Cell[TextData[{ "What you can more safely use, however, is the two-argument form ", StyleBox["ArcTan[x,y]", FontFamily->"Courier"], " to find ", StyleBox["Arg[x+\[ImaginaryI]", FontFamily->"Courier"], StyleBox["\[ThinSpace]", FontFamily->"Courier"], StyleBox["y]", FontFamily->"Courier"], ". The two-argument forms takes account of the quadrant in which the point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " lies. For example:" }], "Text", CellChangeTimes->{{3.4909107160625*^9, 3.4909107974375*^9}, 3.491923537375*^9}], Cell[BoxData[{ RowBox[{"{", RowBox[{ RowBox[{"Arg", "[", RowBox[{ RowBox[{"-", "1"}], "+", "\[ImaginaryI]"}], "]"}], ",", RowBox[{"ArcTan", "[", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "]"}]}], "}"}], "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"Arg", "[", RowBox[{ RowBox[{"-", "1"}], "-", "\[ImaginaryI]"}], "]"}], ",", " ", RowBox[{"ArcTan", "[", RowBox[{ RowBox[{"-", "1"}], ",", RowBox[{"-", "1"}]}], "]"}]}], "}"}]}], "Input", CellChangeTimes->{{3.49091072728125*^9, 3.490910741421875*^9}}], Cell[TextData[{ "And whereas ", StyleBox["ArcTan[1/0]", FontFamily->"Courier"], " is indeterminate, the two-argument form ", StyleBox["ArcTan[0,1]", FontFamily->"Courier"], " works:" }], "Text", CellChangeTimes->{3.49192354546875*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Arg", "[", "\[ImaginaryI]", "]"}], ",", RowBox[{"ArcTan", "[", RowBox[{"0", ",", "1"}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.490910818359375*^9, 3.49091082515625*^9}, 3.491923577140625*^9, 3.49192361746875*^9}], Cell[TextData[{ "But it is still a bit dangerous even to use ", StyleBox["ArcTan[x,y]", FontFamily->"Courier"], " to find ", StyleBox["Arg[x+\[ImaginaryI]y]", FontFamily->"Courier"], ": both ", StyleBox["x", FontFamily->"Courier"], " and ", StyleBox["y", FontFamily->"Courier"], " might be 0. And recall that ", Cell[BoxData[ FormBox[ RowBox[{"Arg", "(", "0", ")"}], TraditionalForm]]], " is not defined mathematically." }], "Text", CellChangeTimes->{{3.49091096384375*^9, 3.4909110041875*^9}, 3.49091105553125*^9}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " What, if anything, does ", StyleBox["Mathematica", FontSlant->"Italic"], " give as the value of ", StyleBox["Arg[0]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490911022609375*^9, 3.490911065609375*^9}}], Cell[TextData[{ "For a way to convert an expression with symbolic variables involving ", StyleBox["Arg", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to one involving ", StyleBox["ArcTan", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", ButtonBox["see below", BaseStyle->"Hyperlink", ButtonData->"ArgToArcTan"], "." }], "Text", CellChangeTimes->{{3.491923629671875*^9, 3.4919236435*^9}, { 3.491924331890625*^9, 3.49192433290625*^9}, {3.49192438340625*^9, 3.49192438846875*^9}, {3.491924483515625*^9, 3.491924547875*^9}, { 3.491924650515625*^9, 3.491924650515625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Traditional notation for modulus (", StyleBox["very", FontSlant->"Italic"], " ", StyleBox["advanced", FontSlant->"Italic"], ")" }], "Subsubsection", ShowGroupOpener->True, CellChangeTimes->{{3.490911310984375*^9, 3.49091131340625*^9}, { 3.491925103765625*^9, 3.4919251044375*^9}}], Cell[TextData[{ "When you are typing text in a text cell, to get the traditional notation ", Cell[BoxData[ FormBox[ RowBox[{"|", "z", "|"}], TraditionalForm]]], " to denote the modulus of a complex number, just use the | key on the \ keykboard." }], "Text"], Cell[TextData[{ "To cause ", StyleBox["Mathematica", FontSlant->"Italic"], " to evaluate the notation ", StyleBox["|z|", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as if it meant ", StyleBox["Abs[z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " takes more work. It probably is more trouble than the result is worth, as \ it requires that you use the ", StyleBox["Notation", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " add-on package that is supplied with ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.49091124628125*^9, 3.490911326828125*^9}, { 3.49091165875*^9, 3.49091169690625*^9}}], Cell[TextData[{ "If you really must use the traditional notation for complex modulus, begin \ by evaluating the following two input cells. The first loads the ", StyleBox["Notation", FontSlant->"Italic"], " package; the second sets up the notational equialence." }], "Text", CellChangeTimes->{{3.49091124628125*^9, 3.490911326828125*^9}, { 3.49091165875*^9, 3.4909117208125*^9}, 3.49192480665625*^9, { 3.491925025640625*^9, 3.491925039859375*^9}}], Cell[BoxData[ RowBox[{"<<", "Notation`"}]], "Input", CellChangeTimes->{{3.4909112101875*^9, 3.490911214828125*^9}, { 3.49192501975*^9, 3.491925022015625*^9}}], Cell[BoxData[ RowBox[{"Notation", "[", RowBox[{ TemplateBox[{RowBox[{"|", "z_", "|"}]}, "NotationTemplateTag"], " ", "\[DoubleLongLeftRightArrow]", " ", TemplateBox[{RowBox[{"Abs", "[", "z_", "]"}]}, "NotationTemplateTag"]}], "]"}]], "Input", CellChangeTimes->{{3.4909114903125*^9, 3.49091150825*^9}, { 3.49091155803125*^9, 3.4909115611875*^9}}], Cell[TextData[{ StyleBox["\[WarningSign]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0]], StyleBox[" Caution:", FontSlant->"Italic"], " Do ", StyleBox["not", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], " type the second of those input cells yourself! Intead, either copy and \ paste the entire cell into your notebook or else reproduce it using the \ Notation palette that pops-up when you load the Notation package." }], "Text", CellChangeTimes->{{3.490911636921875*^9, 3.490911638921875*^9}, { 3.490911729390625*^9, 3.490911736875*^9}}], Cell[TextData[{ "You may now use the traditional notation as input (although ", StyleBox["Mathematica", FontSlant->"Italic"], " will still complain about the syntax by coloring the ", StyleBox["|", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " symbols):" }], "Text", CellChangeTimes->{{3.491924727484375*^9, 3.4919247586875*^9}}], Cell[BoxData[ RowBox[{"|", RowBox[{"2", "+", RowBox[{"3", "I"}]}], "|"}]], "Input"], Cell[TextData[{ "You'll still need to use ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", too, when the expression inside the ", StyleBox["| |", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " involves symbols:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", RowBox[{"|", "z", "|"}], "]"}]}], "Input", CellChangeTimes->{{3.490911759015625*^9, 3.49091177040625*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Converting from Cartesian to polar form", "Subsection", ShowGroupOpener->True, CellChangeTimes->{{3.49091182428125*^9, 3.490911824984375*^9}, 3.4925189045625*^9}], Cell["\<\ Compare the drawings made earlier of the real and imaginary parts of a \ complex number, on the one hand, and of its modulus and principal argument. \ Here they are, side-by-side:\ \>", "Text", CellChangeTimes->{{3.491770865265625*^9, 3.491770916359375*^9}, { 3.491771110203125*^9, 3.4917711188125*^9}}], Cell[BoxData[ RowBox[{"GraphicsRow", "[", RowBox[{"{", RowBox[{"CartesianDrawing", ",", "polarDrawing"}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.49177092040625*^9, 3.491770991296875*^9}, { 3.491911067265625*^9, 3.49191107425*^9}, 3.492518904578125*^9}], Cell["\<\ And here they are again, superimposed:\ \>", "Text", CellChangeTimes->{{3.491771123015625*^9, 3.491771129078125*^9}, 3.49178265821875*^9}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"{", RowBox[{"CartesianDrawing", ",", "polarDrawing"}], "}"}], ",", RowBox[{"PlotLabel", "\[Rule]", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.49177106146875*^9, 3.49177110428125*^9}, 3.491911087828125*^9, 3.492518904578125*^9}], Cell[TextData[{ "These drawings are for one particular complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ". But they suggest, and trigonometry proves, the following general \ relationship:\nIf \n\t", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " with ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " real, \nand if\n\t", Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"|", "z", "|"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "=", RowBox[{"Arg", "(", "z", ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ",\nthen\n\t", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"r", " ", "cos", " ", "\[Theta]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{"r", " ", "sin", " ", "\[Theta]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ",\nand so\n\t", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "r", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". \nIn other words, for a complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], ",\n\t", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"|", "z", "|", RowBox[{ RowBox[{"cos", "(", RowBox[{"Arg", "(", "z", ")"}], ")"}], "+", "\[ImaginaryI]"}], "|", "z", "|", RowBox[{ RowBox[{"sin", "(", RowBox[{"Arg", "(", "z", ")"}], ")"}], "."}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.4914360487925777`*^9, 3.4914361150894527`*^9}, { 3.4914361521832027`*^9, 3.4914361527769527`*^9}, {3.491771137921875*^9, 3.49177140803125*^9}, {3.491771632296875*^9, 3.49177172115625*^9}, 3.49178267465625*^9, {3.49186184475*^9, 3.49186184475*^9}, { 3.49186198853125*^9, 3.49186198853125*^9}, {3.491925508859375*^9, 3.49192568865625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "We call the preceding representation the ", StyleBox["polar form", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of a complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ". Of course it\[CloseCurlyQuote]s just another expression for the ", "Cartesian", " form of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " and so more properly is called the ", StyleBox["trigonometric form", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4914360487925777`*^9, 3.4914361150894527`*^9}, { 3.4914361521832027`*^9, 3.4914361527769527`*^9}, {3.491771137921875*^9, 3.49177140803125*^9}, {3.491771632296875*^9, 3.49177172115625*^9}, { 3.49178267465625*^9, 3.491782676609375*^9}, {3.49192570521875*^9, 3.491925750515625*^9}, 3.4925189046875*^9}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) Form a specific, numeric complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". In ", StyleBox["Mathematica", FontSlant->"Italic"], ", define ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"|", "z", "|"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "=", RowBox[{"Arg", "(", "z", ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", then demonstrate that ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], "+", RowBox[{"\[ImaginaryI]", " ", "r", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ".\n(b) Repeat (a) but now for an arbitrary, symbolic complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". What goes wrong, and why?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49186211228125*^9, 3.49186227384375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "Either of two notations is at times used to denote the polar form:\n\t\ \[FilledSmallCircle] ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"r", " ", "cis", " ", "\[Theta]"}]}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox[ RowBox[{"cis", " ", "\[Theta]"}], TraditionalForm]]], " is just an abbreviation for ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"cos", " ", "\[Theta]"}], "+", RowBox[{"\[ImaginaryI]", " ", "sin", " ", "\[Theta]"}]}], TraditionalForm]]], ".\n\t\[FilledSmallCircle] ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", StyleBox[ TemplateBox[{ "r",StyleBox["\"\[Angle]\"", StripOnInput -> False, FontSize -> Large], "\[Theta]"}, "Row", DisplayFunction->( RowBox[{#, "\[InvisibleSpace]", #2, "\[InvisibleSpace]", #3}]& ), InterpretationFunction->(RowBox[{"Row", "[", RowBox[{"{", RowBox[{#, ",", #2, ",", #3}], "}"}], "]"}]& )], StripOnInput->False, ShowStringCharacters->False]}], TraditionalForm]]], " (see the", ButtonBox[" subsection below", BaseStyle->"Hyperlink", ButtonData->"rTheta"], " about using this notation in ", StyleBox["Mathematica", FontSlant->"Italic"], ")" }], "Text", CellChangeTimes->{{3.49177142359375*^9, 3.491771591359375*^9}, { 3.491771743796875*^9, 3.491771854171875*^9}, {3.491771976140625*^9, 3.491772050265625*^9}, {3.491772290359375*^9, 3.491772298140625*^9}, 3.49177233421875*^9, {3.49177238178125*^9, 3.49177242146875*^9}, { 3.49177246715625*^9, 3.4917724868125*^9}, {3.491782692140625*^9, 3.491782715046875*^9}, {3.49178274778125*^9, 3.49178276828125*^9}, { 3.491862290765625*^9, 3.491862291578125*^9}, {3.49192579453125*^9, 3.491925833171875*^9}, {3.491925864640625*^9, 3.491925865578125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "There is yet another way to express the polar form, by using the complex \ exponential function ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "z"}]], TraditionalForm]]], ". In the case of ", StyleBox["real", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], ", that function satisfies ", StyleBox["Euler\[CloseCurlyQuote]s formula", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ":" }], "Text", CellChangeTimes->{{3.491926167703125*^9, 3.491926299625*^9}, { 3.491926709734375*^9, 3.491926716453125*^9}, {3.49192675659375*^9, 3.491926766875*^9}, {3.4919272994375*^9, 3.491927337484375*^9}, { 3.491927441765625*^9, 3.49192744228125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]], "=", RowBox[{ RowBox[{"cos", " ", "\[Theta]"}], " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\t(", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]], FormatType->"TraditionalForm"], " real)" }], "EmphasisText", CellChangeTimes->{{3.49192739240625*^9, 3.491927435890625*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " already knows that formula:" }], "Text", CellChangeTimes->{{3.491926312453125*^9, 3.4919263195*^9}, { 3.4919267226875*^9, 3.491926724109375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}], "//", "ComplexExpand"}]], "Input", CellChangeTimes->{{3.491926081796875*^9, 3.49192609078125*^9}, { 3.491926324734375*^9, 3.491926331265625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Use Euler\[CloseCurlyQuote]s formula to find the values of ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"\[Pi]", " ", "\[ImaginaryI]"}]], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"(", RowBox[{"\[Pi]", "/", "2"}], ")"}], "\[ImaginaryI]"}]], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"(", RowBox[{"\[DoubledPi]", "/", "4"}], ")"}], "\[ImaginaryI]"}]], TraditionalForm]], FormatType->"TraditionalForm"], ", and ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"2", "\[Pi]", " ", "\[ImaginaryI]"}]], TraditionalForm]], FormatType->"TraditionalForm"], ". Check your answers with ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491926399203125*^9, 3.491926578046875*^9}, {3.491926731765625*^9, 3.49192674271875*^9}}], Cell[TextData[{ "In view of Euler\[CloseCurlyQuote]s formula, the polar form ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "r", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]]], " of a complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " is equivalent to the ", StyleBox["exponential form", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"r", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}]}], TraditionalForm]]], ". This exponential form is itself often referred to as the polar form!" }], "Text", CellChangeTimes->{{3.491926356578125*^9, 3.491926373421875*^9}, { 3.49192674521875*^9, 3.491926748078125*^9}, {3.491926791921875*^9, 3.49192689871875*^9}}], Cell[TextData[{ "To convert a specific, numeric complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " in ", "Cartesian", " form to exponential form ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"r", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}]}], TraditionalForm]]], ", do the obvious thing:" }], "Text", CellChangeTimes->{{3.490911882265625*^9, 3.490911891734375*^9}, { 3.49177183734375*^9, 3.491771838921875*^9}, {3.491926926828125*^9, 3.491926929046875*^9}, 3.49251890484375*^9}], Cell[BoxData[{ RowBox[{ RowBox[{ SubscriptBox["z", "0"], "=", RowBox[{"2", "+", RowBox[{"3", "\[ImaginaryI]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Abs", "[", SubscriptBox["z", "0"], "]"}], " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Arg", "[", SubscriptBox["z", "0"], "]"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.490911897875*^9, 3.490911954578125*^9}, { 3.4917721086875*^9, 3.491772166390625*^9}, {3.491772228796875*^9, 3.491772229921875*^9}, {3.491927465890625*^9, 3.49192746678125*^9}}], Cell[TextData[{ "When the argument is one of the standard values that ", StyleBox["Mathematica", FontSlant->"Italic"], " \"knows\", it will be found explicitly:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ SubscriptBox["z", "1"], "=", RowBox[{"1", "+", "\[ImaginaryI]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Abs", "[", SubscriptBox["z", "1"], "]"}], RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Arg", "[", SubscriptBox["z", "1"], "]"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.490911967765625*^9, 3.490911978359375*^9}, { 3.491927475*^9, 3.4919274755625*^9}}], Cell["\<\ You will ordinarily have considerable difficulty in obtaining the polar form \ of a Cartesian form symbolic expression representing complex numbers. The \ best you will be able to do usually is to obtain an explicit form for the \ modulus. For example:\ \>", "Text", CellChangeTimes->{{3.49091198915625*^9, 3.490912039484375*^9}, 3.49251890496875*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"Abs", "[", "z", "]"}], "]"}], RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Arg", "[", "z", "]"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.49091204246875*^9, 3.490912058703125*^9}}], Cell[TextData[{ "For a more explicit representation of ", StyleBox["Arg[z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " there in terms of ", StyleBox["ArcTan", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", Cell[BoxData[ FormBox[ ButtonBox[ RowBox[{"see", " ", "below"}], BaseStyle->"Hyperlink", ButtonData->"ArgToArcTan"], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.490912396328125*^9, 3.490912397*^9}, 3.491782843609375*^9, {3.491782875703125*^9, 3.49178287653125*^9}, { 3.49186363265625*^9, 3.491863642359375*^9}, {3.4919246225625*^9, 3.4919246225625*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Expressing ", StyleBox["Arg", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " in terms of ", StyleBox["ArcTan", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " (", StyleBox["advanced", FontSlant->"Italic"], ")" }], "Subsubsection", ShowGroupOpener->True, CellChangeTimes->{{3.490912081203125*^9, 3.490912094*^9}, { 3.491862832859375*^9, 3.49186284671875*^9}, {3.491924344109375*^9, 3.49192434628125*^9}}, CellTags->"ArgToArcTan"], Cell[TextData[{ "In the preceding example, the expression ", StyleBox["Arg[z]", FontFamily->"Courier"], " was left in the form ", Cell[BoxData[ RowBox[{"Arg", "[", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}], "]"}]]], ". To obtain a more explicit representation of that, in terms of ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["ArcTan", FontFamily->"Courier"], " function, invoke ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " with the option ", StyleBox["TargetFunctions\[Rule]{Re,Im}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.490912111484375*^9, 3.490912119*^9}, 3.4918633015*^9, { 3.49192429475*^9, 3.491924296609375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";", RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"Arg", "[", "z", "]"}], ",", RowBox[{"TargetFunctions", "\[Rule]", RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}]}]}], "]"}]}]], "Input", CellChangeTimes->{3.491863123125*^9, 3.491863349125*^9}], Cell[TextData[{ "To obtain that representation in the entire example, you may use the ", StyleBox["Mathematica", FontSlant->"Italic"], " \"rule\" ", StyleBox["z:Arg[_]\[RuleDelayed]ComplexExpand[z,TargetFunctions\[Rule]{Re,Im}\ ]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as follows: " }], "Text", CellChangeTimes->{{3.491863530328125*^9, 3.491863553078125*^9}, { 3.4918635904375*^9, 3.491863594875*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", " ", "+", " ", "1"}], ")"}], "2"], " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"2", " ", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"Abs", "[", "z", "]"}], "]"}], " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Arg", "[", "z", "]"}]}], "]"}]}], " ", "/.", " ", RowBox[{ RowBox[{"z", ":", RowBox[{"Arg", "[", "_", "]"}]}], " ", "\[RuleDelayed]", " ", RowBox[{"ComplexExpand", "[", RowBox[{"z", ",", RowBox[{"TargetFunctions", "\[Rule]", RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}]}]}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.4918631461875*^9, 3.491863157609375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Obtain the polar representation of ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "2"], "+", RowBox[{"\[ImaginaryI]", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "-", RowBox[{"x", " ", "y"}]}], ")"}], "2"]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " in a form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], " ", "+", " ", RowBox[{"r", " ", "sin", " ", "\[Theta]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " where ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]], FormatType->"TraditionalForm"], " in turn is expressed in terms of arctan and where ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " is, of course, expressed directly in terms of ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49186367390625*^9, 3.491863813125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "r\[Angle]\[Theta] notation:", StyleBox[" ComplexPolar", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"] }], "Subsubsection", CellChangeTimes->{{3.49091243415625*^9, 3.490912490703125*^9}, { 3.4909125576875*^9, 3.49091259228125*^9}, {3.490912730921875*^9, 3.4909127383125*^9}}, CellTags->"rTheta"], Cell[TextData[{ "The ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ComplexPolar", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " realizes the ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"r", "\[Angle]\[Theta]"}]], TraditionalForm]]], " notation for the polar form, and in a way that you can use in computation." }], "Text", CellChangeTimes->{{3.490912534*^9, 3.49091254925*^9}, {3.49091259853125*^9, 3.4909126861875*^9}, {3.490912834125*^9, 3.4909128436875*^9}, 3.49177253340625*^9, {3.491772662109375*^9, 3.4917726676875*^9}, { 3.491782888453125*^9, 3.491782888984375*^9}}], Cell[BoxData[ RowBox[{"ComplexPolar", "[", RowBox[{ SqrtBox["2"], ",", RowBox[{"\[Pi]", "/", "4"}]}], "]"}]], "Input", CellChangeTimes->{{3.49091293275*^9, 3.490912964921875*^9}, 3.49177267921875*^9}], Cell[TextData[{ "The ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["PolarToComplex", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " converts such a ", StyleBox["ComplexPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " form to ", "Cartesian", " form:" }], "Text", CellChangeTimes->{{3.49178305115625*^9, 3.4917830648125*^9}, { 3.49178309640625*^9, 3.491783115921875*^9}, 3.492518905109375*^9}], Cell[BoxData[ RowBox[{"PolarToComplex", "[", RowBox[{"ComplexPolar", "[", RowBox[{ SqrtBox["2"], ",", RowBox[{"\[Pi]", "/", "4"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.491783120046875*^9, 3.491783130796875*^9}}], Cell[TextData[{ "And the ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ComplexToPolar", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " does the opposite conversion:" }], "Text", CellChangeTimes->{{3.491783136140625*^9, 3.4917831549375*^9}}], Cell[BoxData[ RowBox[{"ComplexToPolar", "[", RowBox[{"1", "+", "\[ImaginaryI]"}], "]"}]], "Input", CellChangeTimes->{{3.49178315671875*^9, 3.49178316378125*^9}}], Cell[TextData[{ "The ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["PolarToExp", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " converts from such a ", StyleBox["ComplexPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " form to exponential form:" }], "Text", CellChangeTimes->{{3.491933053921875*^9, 3.491933092703125*^9}}], Cell[BoxData[ RowBox[{"PolarToExp", "[", RowBox[{"ComplexToPolar", "[", RowBox[{"1", "+", "\[ImaginaryI]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.491933099328125*^9, 3.4919331133125*^9}}], Cell[TextData[{ "Except when the argument is one of the special angles that ", StyleBox["Mathematica", FontSlant->"Italic"], " \[OpenCurlyDoubleQuote]knows\[CloseCurlyDoubleQuote], the best ", StyleBox["ComplexToPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " can do is to give an argument in terms of ", StyleBox["ArcTan", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.491783260765625*^9, 3.491783341890625*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ SubscriptBox["z", "0"], "=", RowBox[{"2", "+", RowBox[{"3", "I"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ComplexToPolar", "[", SubscriptBox["z", "0"], "]"}]}], "Input", CellChangeTimes->{{3.490911897875*^9, 3.490911954578125*^9}, { 3.490912718140625*^9, 3.4909127730625*^9}, {3.490912853765625*^9, 3.49091285896875*^9}, {3.490912940984375*^9, 3.490912941578125*^9}}], Cell[TextData[{ "Although ", StyleBox["ComplexToPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " always expresses the angle in terms of radians, you may enter the second \ argument of ", StyleBox["ComplexPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in degrees instead of radians:" }], "Text", CellChangeTimes->{{3.4919332383125*^9, 3.49193327115625*^9}, { 3.491933381859375*^9, 3.491933399703125*^9}, {3.491933747921875*^9, 3.491933785796875*^9}, {3.49193388015625*^9, 3.491933911125*^9}}], Cell[BoxData[ RowBox[{"ComplexPolar", "[", RowBox[{ SqrtBox["2"], ",", RowBox[{"45", "Degree"}]}], "]"}]], "Input", CellChangeTimes->{{3.491933272578125*^9, 3.491933286296875*^9}, 3.491934031078125*^9}], Cell["\<\ And you may even use a degree symbol, typed as \[EscapeKey]deg\[EscapeKey]\ \[ThinSpace]:\ \>", "Text", CellChangeTimes->{{3.49193340690625*^9, 3.49193344965625*^9}, { 3.491933495765625*^9, 3.491933511796875*^9}}], Cell[BoxData[ RowBox[{"ComplexPolar", "[", RowBox[{ SqrtBox["2"], ",", RowBox[{"45", "\[Degree]"}]}], "]"}]], "Input", CellChangeTimes->{{3.4919334545625*^9, 3.491933483421875*^9}, 3.49193379984375*^9}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Check that the conversion from ", StyleBox["ComplexPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " form to ", "Cartesian", " form also works when you specify the angle in degrees." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491933810625*^9, 3.491933831921875*^9}, {3.49193386309375*^9, 3.49193386684375*^9}, { 3.491933931921875*^9, 3.491933978875*^9}, 3.49251890521875*^9}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Converting from polar to Cartesian form", "Subsection", CellChangeTimes->{{3.49091185084375*^9, 3.490911858078125*^9}, { 3.49191110065625*^9, 3.491911101078125*^9}, 3.492518905328125*^9}], Cell[TextData[{ "To convert a polar form ", Cell[BoxData[ FormBox[ RowBox[{"r", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}], TraditionalForm]]], " to ", "Cartesian", " form\[LongDash]even if that polar form is symbolic\[LongDash]just use ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.491911247953125*^9, 3.491911319375*^9}, { 3.49191153221875*^9, 3.491911538359375*^9}, {3.491911582578125*^9, 3.491911605296875*^9}, 3.4925189054375*^9}], Cell[BoxData[{ RowBox[{ SubscriptBox["z", "2"], "=", RowBox[{"3", RowBox[{"Exp", "[", RowBox[{ FractionBox["\[Pi]", "11"], "\[ImaginaryI]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", SubscriptBox["z", "2"], "]"}]}], "Input", CellChangeTimes->{{3.491911175078125*^9, 3.491911222375*^9}, { 3.491911327234375*^9, 3.491911463171875*^9}, {3.491911493296875*^9, 3.491911549703125*^9}, {3.491927071515625*^9, 3.49192707190625*^9}}], Cell[BoxData[{ RowBox[{"Clear", "[", "z", "]"}], "\[IndentingNewLine]", RowBox[{"z", "=", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", "z", "]"}]}], "Input", CellChangeTimes->{{3.491911175078125*^9, 3.491911222375*^9}, { 3.491911327234375*^9, 3.491911463171875*^9}, {3.491911493296875*^9, 3.4919115548125*^9}}], Cell[TextData[{ "The same method works to convert to ", "Cartesian", " form functions of a complex variable ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " when that variable is already specified in polar form. Here are two \ examples of such conversions." }], "Text", CellChangeTimes->{ 3.491864313375*^9, {3.491911697265625*^9, 3.49191175959375*^9}, 3.492518905546875*^9}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", SuperscriptBox["z", "2"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ SuperscriptBox["z", "2"], "+", RowBox[{"3", "z"}], "+", "\[ImaginaryI]", "+", "5"}], "]"}]], "Input", CellChangeTimes->{{3.49186439778125*^9, 3.491864401703125*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "More about assumed complex real variables and target functions in ", StyleBox["ComplexExpand", FontFamily->"Courier"], " (", StyleBox["advanced", FontSlant->"Italic"], ")" }], "Subsubsection", ShowGroupOpener->True, CellChangeTimes->{{3.490907555859375*^9, 3.490907577453125*^9}}, CellTags->"MoreAdvancedComplexExpand"], Cell[TextData[{ "This continues the discussion begun ", ButtonBox["above", BaseStyle->"Hyperlink", ButtonData:>"AdvancedComplexExpand"], " about ", StyleBox["ComplexExpand", FontFamily->"Courier"], " with a second argument or a ", StyleBox["TargetFunctions", FontFamily->"Courier"], " option." }], "Text"], Cell[TextData[{ "To express a function of a complex variable in terms of that variable's \ modulus and argument, you may use the form of ", StyleBox["ComplexExpand", FontFamily->"Courier"], " in which you declare that variable as complex and specify ", StyleBox["Abs", FontFamily->"Courier"], " and ", StyleBox["Arg", FontFamily->"Courier"], " as the target functions. For example:" }], "Text"], Cell[BoxData[{ RowBox[{"Clear", "[", "z", "]"}], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", RowBox[{ SuperscriptBox["z", "2"], ",", "z", ",", RowBox[{"TargetFunctions", "\[Rule]", RowBox[{"{", RowBox[{"Abs", ",", "Arg"}], "}"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491911817796875*^9, 3.49191182990625*^9}}] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Trig form", "Section", ShowGroupOpener->True], Cell[TextData[{ "Recall that the trigonometric (or \"trig\[CloseCurlyDoubleQuote]) form of a \ complex number ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " is\n\t", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], "+", RowBox[{"\[ImaginaryI]", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]]], " where ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"|", "z", "|"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "=", RowBox[{"Arg", "(", "z", ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\nrather than the equivalent exponential form ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"r", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}]}], TraditionalForm]]], ". 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For example:\ \>", "Text"], Cell[BoxData[ RowBox[{"w", "=", RowBox[{"ExpToTrig", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"1", "/", "3"}]], "]"}]}]], "Input", CellChangeTimes->{{3.491928082078125*^9, 3.4919280839375*^9}, 3.491928594203125*^9}], Cell[TextData[{ "That number is a cube root of ", Cell[BoxData[ FormBox[ RowBox[{"-", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ":" }], "Text", CellChangeTimes->{{3.491928123046875*^9, 3.49192813690625*^9}}], Cell[BoxData[ RowBox[{"Expand", "[", SuperscriptBox["w", "3"], "]"}]], "Input", CellChangeTimes->{{3.491928094671875*^9, 3.4919281063125*^9}, 3.491928602234375*^9}], Cell[TextData[{ "We said \[OpenCurlyDoubleQuote]", StyleBox["a", FontSlant->"Italic"], " cube root\[CloseCurlyDoubleQuote] because ", Cell[BoxData[ FormBox[ RowBox[{"-", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ", like any nonzero complex number, has three different cube roots. 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