(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 20203, 573]*) (*NotebookOutlinePosition[ 26402, 724]*) (* CellTagsIndexPosition[ 25014, 692]*) (*WindowFrame->Normal*) Notebook[{ Cell["Complex Numbers/Complex Plane", "Title"], Cell["Background", "Subtitle"], Cell["The positive square root of -1", "Text", FontWeight->"Plain", CellTags->"mmtag:08:imaginary_numbers__representations_of"], Cell[BoxData[ \(imaginary\ = Sqrt[\(-1\)]\)], "Input"], Cell[BoxData[ \(\((\(-imaginary\))\)^2\)], "Input"], Cell["\<\ Complex numbers are composed of a real part + an imaginary \ part\ \>", "Text", FontWeight->"Plain", CellTags->"mmtag:08:complex_numbers__representations_of"], Cell[BoxData[{ \(\(z1\ = \ a\ + \ \[ImaginaryI]\ b;\)\), "\[IndentingNewLine]", \(\(z2 = \ c\ + \ \[ImaginaryI]\ d;\)\)}], "Input"], Cell["Simple operations on complex numbers", "Text", FontWeight->"Plain"], Cell[BoxData[ \(\(compadd\ = \ z1\ + \ z2;\)\)], "Input"], Cell[BoxData[ \(\(compmult\ = \ z1*z2;\)\)], "Input"], Cell[BoxData[ \(Simplify[compmult, a \[Element] \ Reals\ && \ b \[Element] \ Reals\ && c\ \[Element] \ Reals\ && \ d\ \[Element] \ Reals\ ]\)], "Input", CellTags->{ "mmtag:08:complex_numbers__simplfying", "mmtag:08:Simplify[]__complex_numbers"}], Cell["\<\ Mathematica does not assume that symbols are necessarily \ real...\ \>", "Text", FontWeight->"Plain"], Cell[BoxData[{ \(Re[compadd]\), "\[IndentingNewLine]", \(Im[compadd]\)}], "Input", CellTags->{"mmtag:08:Im[]", "mmtag:08:Re[]"}], Cell[TextData[{ "However, the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["ComplexExpand", FontWeight->"Bold"], " does assume that the variables are real...." }], "Text", FontWeight->"Plain"], Cell[BoxData[ \(ComplexExpand[Re[compadd]]\)], "Input", CellTags->"mmtag:08:ComplexExpand[]"], Cell[BoxData[ \(ComplexExpand[Im[compadd]]\)], "Input"], Cell[BoxData[ \(ComplexExpand[Re[z1/z2]]\)], "Input", CellTags->"mmtag:08:complex_numbers__division"], Cell[BoxData[ \(ComplexExpand[compmult]\)], "Input", CellTags->"mmtag:08:complex_numbers__multiplication"], Cell[BoxData[{ \(ComplexExpand[Re[z1^3]]\), "\[IndentingNewLine]", \(ComplexExpand[Im[z1^3]]\)}], "Input"], Cell["Function to convert to Polar Form", "Subtitle"], Cell[BoxData[ \(Pform[z_]\ := \ Abs[z]\ Exp[\[ImaginaryI]\ Arg[z]]\)], "Input", CellTags->{ "mmtag:08:complex_numbers__polar_form", "mmtag:08:functions__example_of_conversion_of_complex_number_to_polar_\ form"}], Cell[CellGroupData[{ Cell[TextData[{ "Note: the function ", StyleBox["Arg[z]", FontWeight->"Bold"], " returns an angle in the range -\[Pi] to \[Pi] which measures the \ inclination of ", StyleBox["z", FontSlant->"Italic"], " with respect to the +Re axis in the complex plane." }], "Subsubsection"], Cell[BoxData[ \(Pform[z1]\)], "Input"], Cell[BoxData[ \(Pform[z1 /. {a \[Rule] 2, b \[Rule] \(-\[Pi]\)}]\)], "Input"], Cell[BoxData[ \(ComplexExpand[Pform[z1]]\)], "Input"], Cell["Numerical Precision and Complex Rounding", "Subtitle"], Cell[BoxData[ \(ExactlyOne\ = \ Exp[2\ \[Pi]\ \[ImaginaryI]]\)], "Input", CellTags->"mmtag:08:complex_numbers__numerical_precision"], Cell[BoxData[ \(NumericallyOne\ = Exp[N[2\ \[Pi]\ \[ImaginaryI]]]\)], "Input"], Cell[BoxData[ \(Chop[NumericallyOne]\)], "Input", CellTags->"mmtag:08:Chop[]"], Cell[BoxData[ \(Round[NumericallyOne]\)], "Input", CellTags->"mmtag:08:Round[]"], Cell[BoxData[ \(ExactlyI\ = \ Exp[\[Pi]\ \[ImaginaryI]/2]\)], "Input"], Cell[BoxData[ \(NumericallyI\ = \ Exp[N[\[Pi]\ \[ImaginaryI]/2]]\)], "Input"], Cell[BoxData[ \(Round[NumericallyI]\)], "Input"], Cell[BoxData[ \(Chop[NumericallyI]\)], "Input"], Cell[BoxData[ \(ExactlyOnePlusI\ = \ ComplexExpand[\(\@2\) Exp[\[Pi]\ \ \[ImaginaryI]/4]]\)], "Input"], Cell[BoxData[ \(NumericallyOnePlusI\ = \ ComplexExpand[\(\@2\) Exp[N[\[Pi]\ \ \[ImaginaryI]/4]]]\)], "Input"], Cell[BoxData[ \(Chop[NumericallyOnePlusI]\)], "Input"], Cell[BoxData[ \(Round[NumericallyOnePlusI]\)], "Input"], Cell[BoxData[ \(Re[NumericallyOnePlusI]\)], "Input"], Cell[BoxData[ \(Im[NumericallyOnePlusI]\)], "Input"], Cell["Roots to Polynomial Equations", "Subtitle"], Cell["\<\ Complex numbers frequently appear in the solution of roots to \ polynomial equations:\ \>", "Text"], Cell[BoxData[ \(sols\ = \ Solve[\((x^4\ - x^3\ + x\ + 1)\)\ \[Equal] \ 0, x]\)], "Input", CellTags->{ "mmtag:08:Solve[]", "mmtag:08:roots_of_polynomial_equations__example_with_complex_roots"}], Cell["\<\ The next statement produces a list of the complex solutions of the \ polynomial:\ \>", "Text"], Cell[BoxData[ \(x /. sols\)], "Input"], Cell[BoxData[ \(Im[x /. sols]\)], "Input"], Cell[BoxData[ \(ComplexExpand[Im[x /. sols]]\)], "Input"], Cell[BoxData[ \(ComplexExpand[Im[x /. sols]] // N\)], "Input"], Cell[BoxData[ \(ComplexExpand[Re[x /. sols]] // N\)], "Input"], Cell[TextData[{ "Generalize the above to a family of solutions: find ", StyleBox["b", FontSlant->"Italic"], " such that imaginary part of the solution vanishes" }], "Text"], Cell[BoxData[ \(bsols\ = \ Solve[\((x^4\ - x^3\ + b*x\ + 1)\)\ \[Equal] \ 0, x]\)], "Input", CellTags-> "mmtag:08:roots_of_polynomial_equations__example_of_plotting_roots"], Cell["\<\ This will give a list of rules that can be used to find \ solutions;as \"b\" is unspecified the rules depend on the symbol \"b\"\ \>", \ "Text"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(bsols\\)\\\" is similar to existing symbol \ \\\"\\!\\(sols\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \({{x \[Rule] 1\/4 - 1\/2\ \[Sqrt]\((1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + \ b)\)\)\/\((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) + \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\))\) - 1\/2\ \[Sqrt]\((1\/2 - \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 \ + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) - \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\) - \(1 - 8\ b\)\/\ \(4\ \@\(1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 + 9\ b\^2 + \@3\ \ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\) + \((9 + 9\ \ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\)\)\))\)}, {x \[Rule] 1\/4 - 1\/2\ \[Sqrt]\((1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + \ b)\)\)\/\((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) + \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\))\) + 1\/2\ \[Sqrt]\((1\/2 - \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 \ + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) - \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\) - \(1 - 8\ b\)\/\ \(4\ \@\(1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 + 9\ b\^2 + \@3\ \ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\) + \((9 + 9\ \ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\)\)\))\)}, {x \[Rule] 1\/4 + 1\/2\ \[Sqrt]\((1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + \ b)\)\)\/\((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) + \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\))\) - 1\/2\ \[Sqrt]\((1\/2 - \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 \ + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) - \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\) + \(1 - 8\ b\)\/\ \(4\ \@\(1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 + 9\ b\^2 + \@3\ \ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\) + \((9 + 9\ \ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\)\)\))\)}, {x \[Rule] 1\/4 + 1\/2\ \[Sqrt]\((1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + \ b)\)\)\/\((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) + \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\))\) + 1\/2\ \[Sqrt]\((1\/2 - \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 \ + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\) - \((9 + 9\ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - \ 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\) + \(1 - 8\ b\)\/\ \(4\ \@\(1\/4 + \(\((2\/3)\)\^\(1/3\)\ \((4 + b)\)\)\/\((9 + 9\ b\^2 + \@3\ \ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ b\^4\))\)\^\(1/3\) + \((9 + 9\ \ b\^2 + \@3\ \@\(\(-229\) - 192\ b + 6\ b\^2 - 4\ b\^3 + 27\ \ b\^4\))\)\^\(1/3\)\/\(2\^\(1/3\)\ 3\^\(2/3\)\)\)\))\)}}\)], "Output"], Cell["\<\ Because it is a long set of rules and hard to follow, let's look at \ the form of bsols:\ \>", "Text"], Cell[BoxData[ \(Dimensions[bsols]\)], "Input", CellTags->"mmtag:08:Dimensions[]"], Cell[TextData[{ StyleBox["Short", FontWeight->"Bold"], " produces a ", StyleBox["very", FontSlant->"Italic"], " abbreviated form of the solution\[Ellipsis] in this case limited to 3 \ lines by the optional parameter." }], "Text"], Cell[BoxData[ \(Short[bsols, 3]\)], "Input", CellTags->"mmtag:08:Short[]"], Cell[TextData[{ "So we see that ", StyleBox["bsols", FontSlant->"Italic"], " is a list of length 4 of list containing one rule. (Solutions to \ equations are always this way, it is a list of the number of solutions, each \ member being a rule for each variable that is solved for...)" }], "Text"], Cell[TextData[{ "In our case of one variable, the extra layer of lists is not terribly \ useful, one way to get rid of the extra layers is to use ", StyleBox["Flatten", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[{ \(Dimensions[Flatten[bsols]]\), "\[IndentingNewLine]", \(Short[Flatten[bsols], 3]\)}], "Input", CellTags->"mmtag:08:Flatten[]"], Cell[TextData[{ "In the next command, we produce a list of values (not of rules, because we \ have taken ", StyleBox["x", FontSlant->"Italic"], " and applied every rule in ", StyleBox["bsols", FontSlant->"Italic"], " to it... These values are the imaginary parts of the solutions ", StyleBox["x", FontSlant->"Italic"], " that make the polynomial vanish (and a function of ", StyleBox["b", FontSlant->"Italic"], ", because it hasn't been specified yet)" }], "Text"], Cell[BoxData[ \(\(imb\ = ComplexExpand[Im[x /. bsols]];\)\)], "Input"], Cell[BoxData[{ \(Dimensions[imb]\), "\[IndentingNewLine]", \(Short[imb[\([1]\)]]\)}], "Input"], Cell[TextData[{ "And likewise for the real parts of ", StyleBox["x", FontSlant->"Italic"], " that solve the polynomial equation" }], "Text"], Cell[BoxData[ \(\(imr = ComplexExpand[Re[x /. bsols]];\)\)], "Input"], Cell[TextData[{ StyleBox["Plot[]", FontWeight->"Bold"], " expects a list of functions to plot, The \"", StyleBox["Flatten", FontWeight->"Bold"], "\" function ensures that ", StyleBox["Plot", FontWeight->"Bold"], " is getting a list, but it is redundant in this particular case. This \ plot works as follows, for each member in the list, plot the result of \ replacing ", StyleBox["b", FontSlant->"Italic"], " with ", StyleBox["q", FontSlant->"Italic"], " for values of ", StyleBox["q", FontSlant->"Italic"], " between -10, 10---it is a long-handed way of seeing exactly what is going \ on and demonstrates the replacement technique.\n\nSo the following should be \ a plot of the imaginary values of ", StyleBox["x", FontSlant->"Italic"], " as a function of ", StyleBox["b", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ \(Plot[ Evaluate[Flatten[imb] /. b \[Rule] q], {q, \(-10\), 10}]\)], "Input", CellTags->{ "mmtag:08:plots__examples_of_modifying", "mmtag:08:plots__examples_of_effect_of_numerical_noise"}], Cell["\<\ In fact, we should be able to do the same thing without \ replacement as the following shows, it is probably instructive to show both \ ways.\ \>", "Text"], Cell[BoxData[ \(Plot[Evaluate[imb], {b, \(-10\), 10}]\)], "Input"], Cell[TextData[{ "There are a few problems that make it difficult to interpret the \ graph---one is the numerical noise that makes the solutions\njump back and \ forth; second, because all the colors are the same, it is not clear which \ values of ", StyleBox["x", FontSlant->"Italic"], " belong to the same solution. \n\nLet's first try to make each member of \ the list (remember, there are 4 because it is a fourth-order polynomial and \ because ", StyleBox["Dimensions[imb]", FontWeight->"Bold"], " told us so..." }], "Text"], Cell[BoxData[ \(Plot[Evaluate[Flatten[imb] /. b \[Rule] q], {q, \(-10\), 10}, PlotStyle \[Rule] Table[{Hue[1 - a/6]}, {a, 1, 4}]]\)], "Input", CellTags->"mmtag:08:plots__examples_of_adding_color"], Cell[TextData[{ "The plot above is a little better, it looks like the blue curve comes in \ from the northeast and then then its imaginary part vanishes at a critical \ values of ", StyleBox["q", FontSlant->"Italic"], " (around -0.5), the cyan curve is probably the minus values of the blue \ curve... and the same thing for yellow and green. It is much easier to see \ the branches of solutions for the real parts below." }], "Text"], Cell[BoxData[ \(Plot[Evaluate[Flatten[imr] /. b \[Rule] q], {q, \(-10\), 10}, PlotStyle \[Rule] Table[{Hue[1 - a/6]}, {a, 1, 4}]]\)], "Input"], Cell["\<\ But here, because the lines are the same thickness, we don't know \ if the cyan and blue curves just \"stop.\" Let's find out by also adjusting \ their thickness.\ \>", "Text"], Cell[BoxData[ RowBox[{\(XML`MathML`BoxesToMathML::"notboxes"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\({\\(\\(RowBox[\\(\\({\\(\\(RowBox[\\(\\({\\\"z1\\\", \\\" \\\", \\\ \"=\\\", \\\" \\\", \\(\\(RowBox[\\(\\({\\\"a\\\", \\\" \\\", \\\"+\\\", \\\" \ \\\", \\(\\(RowBox[\\(\\(\[LeftSkeleton] 1 \ \[RightSkeleton]\\)\\)]\\)\\)}\\)\\)]\\)\\)}\\)\\)]\\)\\), \ \\\";\\\"}\\)\\)]\\)\\), \\(\\(RowBox[\\(\\({\\(\\(\[LeftSkeleton] 1 \ \[RightSkeleton]\\)\\), \\\";\\\"}\\)\\)]\\)\\)}\\) is not a valid box \ structure. The first argument in \\!\\(XML`MathML`BoxesToMathML[\\(\\(\ \[LeftSkeleton] 1 \[RightSkeleton]\\)\\)]\\) must be a valid box structure. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"XML`MathML`BoxesToMathML::notboxes\\\ \"]\\)\"\>"}]], "Message"], Cell[BoxData[ RowBox[{\(XML`MathML`BoxesToMathML::"notboxes"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\({\\(\\(RowBox[\\(\\({\\\"Re\\\", \\\"[\\\", \\\"compadd\\\", \\\"]\ \\\"}\\)\\)]\\)\\), \\(\\(RowBox[\\(\\({\\\"Im\\\", \\\"[\\\", \ \\\"compadd\\\", \\\"]\\\"}\\)\\)]\\)\\)}\\) is not a valid box structure. \ The first argument in \ \\!\\(XML`MathML`BoxesToMathML[\\(\\(\\(\\({\\(\\(RowBox[\\(\\({\\\"Re\\\", \ \\\"[\\\", \\\"compadd\\\", \\\"]\\\"}\\)\\)]\\)\\), \ \\(\\(RowBox[\\(\\({\\\"Im\\\", \\\"[\\\", \\\"compadd\\\", \\\"]\\\"}\\)\\)]\ \\)\\)}\\)\\), \\(\\(\[LeftSkeleton] 2 \[RightSkeleton]\\)\\), \\(\\(\\(\\(\\\ \"\\\"\[Ellipsis]\\\"\\\"\\)\\) \[Rule] \ \\(\\(\\\"\\\"\[Ellipsis]\\\"\\\"\\)\\)\\)\\)\\)\\)]\\) must be a valid box \ structure. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"XML`MathML`BoxesToMathML::notboxes\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ RowBox[{\(XML`MathML`BoxesToMathML::"notboxes"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\({\\(\\(RowBox[\\(\\({\\\"ComplexExpand\\\", \\\"[\\\", \ \\(\\(RowBox[\\(\\({\\\"Re\\\", \\\"[\\\", \\(\\(RowBox[\\(\\({\\\"z1\\\", \\\ \"^\\\", \\\"3\\\"}\\)\\)]\\)\\), \\\"]\\\"}\\)\\)]\\)\\), \ \\\"]\\\"}\\)\\)]\\)\\), \\(\\(RowBox[\\(\\({\\(\[LeftSkeleton] 1 \ \[RightSkeleton]\\)}\\)\\)]\\)\\)}\\) is not a valid box structure. The first \ argument in \ \\!\\(XML`MathML`BoxesToMathML[\\(\\(\\(\\({\\(\\(RowBox[\\(\\({\\\"\ ComplexExpand\\\", \\\"[\\\", \\(\\(RowBox[\\(\\(\[LeftSkeleton] 1 \ \[RightSkeleton]\\)\\)]\\)\\), \\\"]\\\"}\\)\\)]\\)\\), \ \\(\\(RowBox[\\(\\({\\(\[LeftSkeleton] 1 \ \[RightSkeleton]\\)}\\)\\)]\\)\\)}\\)\\), \\(\\(\[LeftSkeleton] 2 \ \[RightSkeleton]\\)\\), \\(\\(\\\"Entities\\\" \[Rule] \\(\\(\\\"M\\\"\ \[Ellipsis]\\\"L\\\"\\)\\)\\)\\)\\)\\)]\\) must be a valid box structure. \\!\ \\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"XML`MathML`BoxesToMathML::notboxes\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output of \ \\!\\(XML`MathML`BoxesToMathML :: \\\"notboxes\\\"\\) will be suppressed \ during this calculation. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::stop\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \(Plot[Evaluate[Flatten[imr] /. b \[Rule] q], {q, \(-10\), 10}, PlotStyle \[Rule] Table[{Hue[1 - a/6], Thickness[0.05 - .01*a]}, {a, 1, 4}]]\)], "Input", CellTags->"mmtag:08:plots__examples_of_changing_line_thicknesses"], Cell[TextData[{ "It is pretty clear that the parameter ", StyleBox["b", FontSlant->"Italic"], " is behaving like a \"pitchfork\" bifurcation---there is one value of ", StyleBox["x", FontSlant->"Italic"], " upto a critical value of ", StyleBox["b", FontSlant->"Italic"], ", where ", StyleBox["x", FontSlant->"Italic"], " splits into two solutions. This is a picture of two isolated \ pictchforks." }], "Text"] }, Open ]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{45, 1920}, {0, 1178}}, ScreenStyleEnvironment->"Presentation", WindowSize->{1251, 880}, WindowMargins->{{110, Automatic}, {Automatic, 0}}, WindowTitle->"Lecture 08 MIT 3.016 (Fall 2005) \[Copyright] W. Craig Carter \ 2003-2004", ShowCellLabel->False, CellLabelAutoDelete->True, StyleDefinitions -> "3016_Carter.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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