(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 25851, 825] NotebookOptionsPosition[ 16728, 588] NotebookOutlinePosition[ 21972, 722] CellTagsIndexPosition[ 20778, 694] WindowTitle->Lecture 08 MIT 3.016 (Fall 2009) \251 W. Craig Carter \ 2003--2009 WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Complex Numbers/Complex Plane", "Title"], Cell[CellGroupData[{ Cell["Background", "Subtitle"], Cell["The positive square root of -1", "Text", FontWeight->"Plain", CellTags->"mmtag:08:imaginary_numbers__representations_of"], Cell[BoxData[ RowBox[{"imaginary", " ", "=", RowBox[{"Sqrt", "[", RowBox[{"-", "1"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"-", "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[{ RowBox[{ RowBox[{"z1", " ", "=", " ", RowBox[{"a", " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "b"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"z2", "=", " ", RowBox[{"c", " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "d"}]}]}], ";"}]}], "Input"], Cell["Simple operations on complex numbers", "Text", FontWeight->"Plain"], Cell[BoxData[ RowBox[{ RowBox[{"compadd", " ", "=", " ", RowBox[{"z1", " ", "+", " ", "z2"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"compmult", " ", "=", " ", RowBox[{"z1", "*", "z2"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"compmult", ",", RowBox[{ RowBox[{"a", "\[Element]", " ", "Reals"}], " ", "&&", " ", RowBox[{"b", "\[Element]", " ", "Reals"}], " ", "&&", RowBox[{"c", " ", "\[Element]", " ", "Reals"}], " ", "&&", " ", RowBox[{"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[{ RowBox[{"Re", "[", "compadd", "]"}], "\[IndentingNewLine]", RowBox[{"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[ RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", "compadd", "]"}], "]"}]], "Input", CellTags->"mmtag:08:ComplexExpand[]"], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"Im", "[", "compadd", "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", RowBox[{"z1", "/", "z2"}], "]"}], "]"}]], "Input", CellTags->"mmtag:08:complex_numbers__division"], Cell[BoxData[ RowBox[{"ComplexExpand", "[", "compmult", "]"}]], "Input", CellTags->"mmtag:08:complex_numbers__multiplication"], Cell[BoxData[{ RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", RowBox[{"z1", "^", "3"}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", RowBox[{"Im", "[", RowBox[{"z1", "^", "3"}], "]"}], "]"}]}], "Input"], Cell["Function to convert to Polar Form", "Section"], Cell[BoxData[ RowBox[{ RowBox[{"Pform", "[", "z_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Abs", "[", "z", "]"}], " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Arg", "[", "z", "]"}]}], "]"}]}]}]], "Input", CellTags->{ "mmtag:08:complex_numbers__polar_form", "mmtag:08:functions__example_of_conversion_of_complex_number_to_polar_\ form"}], 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[ RowBox[{"Pform", "[", "z1", "]"}]], "Input"], Cell[BoxData[ RowBox[{"Pform", "[", RowBox[{"z1", "/.", RowBox[{"{", RowBox[{ RowBox[{"a", "\[Rule]", "2"}], ",", RowBox[{"b", "\[Rule]", RowBox[{"-", "\[Pi]"}]}]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"Pform", "[", "z1", "]"}], "]"}]], "Input"] }, Open ]], 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Cell[BoxData[ RowBox[{"Im", "[", "NumericallyOnePlusI", "]"}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Complex Valued Roots to Polynomial Equations", "Subtitle"], Cell["\<\ Complex numbers frequently appear in the solution of roots to polynomial \ equations:\ \>", "Text"], Cell[BoxData[ RowBox[{"sols", " ", "=", " ", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "^", "4"}], " ", "-", RowBox[{"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[ RowBox[{"x", "/.", "sols"}]], "Input"], Cell[BoxData[ RowBox[{"Im", "[", RowBox[{"x", "/.", "sols"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"Im", "[", RowBox[{"x", "/.", "sols"}], "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"Im", "[", RowBox[{"x", "/.", "sols"}], "]"}], "]"}], "//", "N"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", RowBox[{"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[ RowBox[{"bsols", " ", "=", " ", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "^", "4"}], " ", "-", RowBox[{"x", "^", "3"}], " ", "+", RowBox[{"b", "*", "x"}], " ", "+", "1"}], ")"}], " ", "\[Equal]", " ", "0"}], ",", "x"}], "]"}]}]], "Input", CellTags-> "mmtag:08:roots_of_polynomial_equations__example_of_plotting_roots"], Cell[TextData[{ "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\"", "\n", "Because it is a long set of rules and hard to follow, let's look at the \ form of bsols:", "\n", 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", CellChangeTimes->{3.398178369615646*^9}], Cell[BoxData[{ RowBox[{"Dimensions", "[", "bsols", "]"}], "\[IndentingNewLine]", RowBox[{"Short", "[", RowBox[{"bsols", ",", "4"}], "]"}]}], "Input", CellChangeTimes->{{3.398178350027873*^9, 3.398178351680067*^9}}, 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[{ RowBox[{"Dimensions", "[", RowBox[{"Flatten", "[", "bsols", "]"}], "]"}], "\[IndentingNewLine]", RowBox[{"Short", "[", RowBox[{ RowBox[{"Flatten", "[", "bsols", "]"}], ",", "4"}], "]"}]}], "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["\<\ **Note,on some machines this next command will take a while to finish**\ \>", "Subsubtitle"], Cell[BoxData[ RowBox[{ RowBox[{"SolsbImag", "=", RowBox[{"ComplexExpand", "[", RowBox[{"Im", "[", RowBox[{"x", "/.", "bsols"}], "]"}], "]"}]}], ";"}]], "Input"], Cell[BoxData[{ RowBox[{"Dimensions", "[", "SolsbImag", "]"}], "\[IndentingNewLine]", RowBox[{"Short", "[", RowBox[{"SolsbImag", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "Input"], Cell[TextData[{ "And likewise for the real parts of ", StyleBox["x", FontSlant->"Italic"], " that solve the polynomial equation" }], "Text"], Cell["\<\ **Note,on some machines this next command will take a while to finish**\ \>", "Subsubtitle"], Cell[BoxData[ RowBox[{ RowBox[{"SolsbReal", "=", RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", RowBox[{"x", "/.", "bsols"}], "]"}], "]"}]}], ";"}]], "Input"], Cell[TextData[{ "This plot works as follows, for each member in the list, plot the result of \ replacing ", StyleBox["b", FontSlant->"Italic"], " with values between -10, 10. So 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[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", "SolsbImag", "]"}], ",", RowBox[{"{", RowBox[{"b", ",", RowBox[{"-", "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 jump 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", CellChangeTimes->{{3.394611730110013*^9, 3.394611731211618*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", "SolsbImag", "]"}], ",", RowBox[{"{", RowBox[{"b", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"Hue", "[", RowBox[{"1", "-", RowBox[{"a", "/", "6"}]}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"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["b ", 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[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", "SolsbReal", "]"}], ",", RowBox[{"{", RowBox[{"b", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"Hue", "[", RowBox[{"1", "-", RowBox[{"a", "/", "6"}]}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"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[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", "SolsbReal", "]"}], ",", RowBox[{"{", RowBox[{"b", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Hue", "[", RowBox[{"1", "-", RowBox[{"a", "/", "6"}]}], "]"}], ",", RowBox[{"Thickness", "[", RowBox[{"0.05", "-", RowBox[{".01", "*", "a"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"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. 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