(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 8910, 263] NotebookOptionsPosition[ 8322, 239] NotebookOutlinePosition[ 8666, 254] CellTagsIndexPosition[ 8623, 251] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"$Assumptions", " ", "=", " ", RowBox[{"x", " ", "\[Element]", " ", "Reals"}]}], " ", ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"(*", RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"x", "^", "2"}]}], ")"}], "^", RowBox[{"(", RowBox[{"1", "/", "4"}], ")"}]}], "*)"}], " ", RowBox[{ RowBox[{"Cos", "[", RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}]}], " ", RowBox[{"ArcTan", "[", " ", "x", " ", "]"}]}], " ", "]"}], " ", "//", " ", "Simplify"}]}]}], "Input", CellChangeTimes->{{3.6869226792097435`*^9, 3.6869228353424664`*^9}}], Cell[BoxData[ RowBox[{"Cos", "[", FractionBox[ RowBox[{"ArcTan", "[", "x", "]"}], "2"], "]"}]], "Output", CellChangeTimes->{{3.6869227337732344`*^9, 3.6869227961089983`*^9}, { 3.686922827421666*^9, 3.686922835967491*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"TrigToExp", "[", RowBox[{"Cos", "[", FractionBox[ RowBox[{"ArcTan", "[", "x", "]"}], "2"], "]"}], "]"}], " ", "//", " ", "Simplify"}], ")"}], " ", "+", " ", RowBox[{"I", RowBox[{"(", RowBox[{ RowBox[{"TrigToExp", "[", RowBox[{"-", RowBox[{"Sin", "[", FractionBox[ RowBox[{"ArcTan", "[", "x", "]"}], "2"], "]"}]}], "]"}], " ", "//", " ", "Simplify"}], ")"}]}]}], " ", "//", " ", "Simplify"}], ")"}], "^", "2"}], "*)"}], "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{{3.6869228801367636`*^9, 3.6869229034969554`*^9}, { 3.686922942836265*^9, 3.6869230519577627`*^9}, 3.686926040732066*^9}, NumberMarks->False], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"ClearAll", "[", RowBox[{ "t", ",", "f", ",", "sigma", ",", " ", "epsilon", ",", " ", "omega", ",", " ", "mu", ",", " ", "epsilon", ",", " ", "alpha", ",", " ", "beta"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"t", " ", "=", " ", RowBox[{"sigma", "/", RowBox[{"(", RowBox[{"epsilon", " ", "omega"}], ")"}]}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "=", " ", RowBox[{"Sqrt", "[", " ", RowBox[{ RowBox[{"omega", "^", "2"}], " ", "mu", " ", "epsilon"}], " ", "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"beta", " ", "=", " ", RowBox[{"f", " ", RowBox[{ RowBox[{"Sqrt", "[", " ", RowBox[{"1", " ", "+", " ", RowBox[{"Sqrt", "[", " ", RowBox[{"1", " ", "+", " ", RowBox[{"t", "^", "2"}]}], "]"}]}], "]"}], "/", RowBox[{"Sqrt", "[", "2", "]"}]}]}]}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"alpha", "=", " ", RowBox[{"f", " ", RowBox[{"t", "/", RowBox[{"(", RowBox[{ RowBox[{"Sqrt", "[", "2", "]"}], RowBox[{"Sqrt", "[", " ", RowBox[{"1", " ", "+", " ", RowBox[{"Sqrt", "[", " ", RowBox[{"1", " ", "+", " ", RowBox[{"t", "^", "2"}]}], "]"}]}], "]"}]}], ")"}]}]}]}], " ", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"alpha", " ", "+", " ", RowBox[{"I", " ", "beta"}]}], ")"}], "^", "2"}], " ", "//", " ", "FullSimplify"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.686926055618533*^9, 3.6869261649066916`*^9}, { 3.686926266000779*^9, 3.686926268439103*^9}, {3.6869264886652822`*^9, 3.6869264993574467`*^9}, {3.686926570002511*^9, 3.686926600807607*^9}, { 3.686926631940881*^9, 3.6869267035191975`*^9}, {3.686927255289835*^9, 3.6869272807035737`*^9}, {3.686927345146467*^9, 3.686927427117852*^9}, { 3.68692763691043*^9, 3.6869277415463843`*^9}, {3.6869277721396627`*^9, 3.6869278023961997`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SqrtBox[ RowBox[{"epsilon", " ", "mu", " ", SuperscriptBox["omega", "2"]}]], " ", SqrtBox[ RowBox[{"1", "+", SqrtBox[ RowBox[{"1", "+", FractionBox[ SuperscriptBox["sigma", "2"], RowBox[{ SuperscriptBox["epsilon", "2"], " ", SuperscriptBox["omega", "2"]}]]}]]}]]}], SqrtBox["2"]]], "Output", CellChangeTimes->{{3.6869261364142733`*^9, 3.6869261657348433`*^9}, 3.6869262693144255`*^9, 3.6869265002637353`*^9, {3.6869265797084656`*^9, 3.686926601495097*^9}, {3.6869266359570446`*^9, 3.6869266731252427`*^9}, 3.686926704472678*^9, 3.686927364925233*^9, 3.686927427664693*^9, { 3.686927697600586*^9, 3.6869277425464263`*^9}, {3.686927792772772*^9, 3.6869278028499002`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SqrtBox[ RowBox[{"epsilon", " ", "mu", " ", SuperscriptBox["omega", "2"]}]], " ", "sigma"}], RowBox[{ SqrtBox["2"], " ", "epsilon", " ", "omega", " ", SqrtBox[ RowBox[{"1", "+", SqrtBox[ RowBox[{"1", "+", FractionBox[ SuperscriptBox["sigma", "2"], RowBox[{ SuperscriptBox["epsilon", "2"], " ", SuperscriptBox["omega", "2"]}]]}]]}]]}]]], "Output", CellChangeTimes->{{3.6869261364142733`*^9, 3.6869261657348433`*^9}, 3.6869262693144255`*^9, 3.6869265002637353`*^9, {3.6869265797084656`*^9, 3.686926601495097*^9}, {3.6869266359570446`*^9, 3.6869266731252427`*^9}, 3.686926704472678*^9, 3.686927364925233*^9, 3.686927427664693*^9, { 3.686927697600586*^9, 3.6869277425464263`*^9}, {3.686927792772772*^9, 3.6869278028499002`*^9}}], Cell[BoxData[ RowBox[{"mu", " ", "omega", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "epsilon"}], " ", "omega"}], "+", RowBox[{"\[ImaginaryI]", " ", "sigma"}]}], ")"}]}]], "Output", CellChangeTimes->{{3.6869261364142733`*^9, 3.6869261657348433`*^9}, 3.6869262693144255`*^9, 3.6869265002637353`*^9, {3.6869265797084656`*^9, 3.686926601495097*^9}, {3.6869266359570446`*^9, 3.6869266731252427`*^9}, 3.686926704472678*^9, 3.686927364925233*^9, 3.686927427664693*^9, { 3.686927697600586*^9, 3.6869277425464263`*^9}, {3.686927792772772*^9, 3.6869278028655252`*^9}}] }, Open ]], Cell[BoxData[{ RowBox[{ RowBox[{"ClearAll", "[", "format", "]"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ RowBox[{"format", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"omega", " ", "\[Rule]", " ", "\[Omega]"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"sigma", " ", "\[Rule]", " ", SubscriptBox["\[Sigma]", "0"]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"mu", " ", "\[Rule]", " ", "\[Mu]"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"epsilon", " ", "\[Rule]", " ", SuperscriptBox["\[Epsilon]", "'"]}], "}"}]}], "}"}]}], " ", ";"}], "*)"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"alpha", " ", "/.", " ", RowBox[{"omega", " ", "\[Rule]", " ", "\[Omega]"}]}], ")"}], " ", "/.", " ", RowBox[{"sigma", " ", "\[Rule]", " ", SubscriptBox["\[Sigma]", "0"]}]}], ")"}], " ", "/.", " ", RowBox[{"mu", " ", "\[Rule]", " ", "\[Mu]"}]}], " ", ")"}], "/.", " ", RowBox[{"epsilon", " ", "\[Rule]", " ", SuperscriptBox["\[Epsilon]", "'"]}]}]}], "Input", CellChangeTimes->{{3.68692775627722*^9, 3.686927758684225*^9}, { 3.686927824019102*^9, 3.68692784990564*^9}, {3.6869279438788157`*^9, 3.6869280246876583`*^9}, {3.686928099866703*^9, 3.6869282681163316`*^9}}] }, WindowSize->{707, 636}, WindowMargins->{{49, Automatic}, {Automatic, 62}}, FrontEndVersion->"9.0 for Microsoft Windows (64-bit) (January 25, 2013)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 709, 22, 116, "Input"], Cell[1291, 46, 231, 5, 58, "Output"] }, Open ]], Cell[1537, 54, 926, 26, 154, "Input"], Cell[CellGroupData[{ Cell[2488, 84, 2058, 53, 241, "Input"], Cell[4549, 139, 808, 21, 115, "Output"], Cell[5360, 162, 866, 22, 156, "Output"], Cell[6229, 186, 607, 12, 40, "Output"] }, Open ]], Cell[6851, 201, 1467, 36, 200, "Input"] } ] *) (* End of internal cache information *)