(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 69505, 2339] NotebookOptionsPosition[ 60658, 2105] NotebookOutlinePosition[ 61039, 2122] CellTagsIndexPosition[ 60996, 2119] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Log-Normal Distribution", "Title",ExpressionUUID->"39cd4b35-a55f-4709-bc06-2803e625c554"], Cell[CellGroupData[{ Cell["Author", "Subsection",ExpressionUUID->"476dc8de-b39e-481a-9ae6-c7e82b1f70e3"], Cell["\", "Text",ExpressionUUID->"3fef5869-824d-4ee0-983c-3320c93efb3a"], Cell[TextData[{ "This notebook downloaded from ", ButtonBox["http://mathworld.wolfram.com/notebooks/Statistics/\ LogNormalDistribution.nb", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/notebooks/Statistics/\ LogNormalDistribution.nb"], None}], "." }], "Text",ExpressionUUID->"0391f0e6-e6fc-4813-98f5-e5c3dd63044d"], Cell[TextData[{ "For more information, see Eric's ", StyleBox["MathWorld", FontSlant->"Italic"], " entry ", ButtonBox["http://mathworld.wolfram.com/LogNormalDistribution.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/LogNormalDistribution.html"], None}], "." }], "Text",ExpressionUUID->"7a691740-82cf-4853-ab51-eb596309e6eb"], Cell["\", "Text",ExpressionUUID->"3478df57-8d99-4c52-9a52-d96423a2f325"] }, Open ]], Cell[CellGroupData[{ Cell["Plots", "Section",ExpressionUUID->"1586a345-0bd2-4902-a70e-c172014c6101"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"GraphicsArray", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"#", "[", RowBox[{ RowBox[{"LogNormalDistribution", "[", RowBox[{"1", ",", "1"}], "]"}], ",", "x"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", ".01", ",", "20"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"PlotStyle", "\[Rule]", "Red"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"TraditionalForm", "/@", RowBox[{"{", RowBox[{"x", ",", RowBox[{"#", "[", "x", "]"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", "None"}], ",", RowBox[{"DisplayFunction", "\[Rule]", "Identity"}]}], "]"}], "&"}], "/@", RowBox[{"{", RowBox[{"PDF", ",", "CDF"}], "}"}]}], "]"}], ",", RowBox[{"ImageSize", "\[Rule]", "500"}]}], "\[IndentingNewLine]", "]"}]], "Input",ExpressionUUID->"f7ef3eed-becb-49b5-a221-8cde03e52026"], Cell[GraphicsData["PostScript", "\"], "Graphics", ImageSize->{500, 147.125},ImageCache->GraphicsData["CompressedBitmap", "\"],ImageRangeCache->{{{0., 499.}, {146.125, 0.}} -> {0., 0., 0., 0.}, \ {{13.0625, 238.188}, {142.625, 3.4375}} -> {-3., 0., 0., 0.}, {{260.75, \ 485.875}, {142.625, 3.4375}} -> {-30., 0., 0., \ 0.}},ExpressionUUID->"9f4c6d55-c8a5-4a16-ab20-d34ef607eca0"], Cell[BoxData[ TagBox[ RowBox[{"\[SkeletonIndicator]", "GraphicsArray", "\[SkeletonIndicator]"}], False, Editable->False]], "Output",ExpressionUUID->"7a9cd686-163c-4e86-a3d8-\ 6a8c4b37379b"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Stats", "Section",ExpressionUUID->"edea9a89-6bac-4ef3-9f52-fb0b34c25c24"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"Stats", "[", RowBox[{ RowBox[{"LogNormalDistribution", "[", RowBox[{"M", ",", "S"}], "]"}], ",", "x"}], "]"}], "//", "FullSimplify"}], "//", "StatsForm"}], "//", "TraditionalForm"}]], "Input",\ ExpressionUUID->"52a1fe5f-46b6-4c70-b6ac-21de9f297f09"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ TagBox[GridBox[{ {"Domain", TagBox[ RowBox[{"Interval", "[", RowBox[{"{", RowBox[{"0", ",", "\[Infinity]"}], "}"}], "]"}], HoldForm]}, { RowBox[{"P", "(", "x", ")"}], FractionBox[ SuperscriptBox[ TagBox["e", Function[{}, E]], RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", TagBox[ RowBox[{"ln", TagBox["x", Log, Editable->True]}], InterpretTemplate[ Function[BoxForm`e$, Log[BoxForm`e$]]]]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]]}, { RowBox[{"D", "(", "x", ")"}], RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"erf", "(", FractionBox[ RowBox[{ TagBox[ RowBox[{"ln", TagBox["x", Log, Editable->True]}], InterpretTemplate[ Function[BoxForm`e$, Log[BoxForm`e$]]]], "-", "M"}], RowBox[{ SqrtBox["2"], " ", "S"}]], ")"}], "+", "1"}], ")"}]}]}, {"\[Mu]", SuperscriptBox[ TagBox["e", Function[{}, E]], RowBox[{ FractionBox[ SuperscriptBox["S", "2"], "2"], "+", "M"}]]}, { SuperscriptBox["\[Sigma]", "2"], RowBox[{ SuperscriptBox[ TagBox["e", Function[{}, E]], RowBox[{ SuperscriptBox["S", "2"], "+", RowBox[{"2", " ", "M"}]}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox[ TagBox["e", Function[{}, E]], SuperscriptBox["S", "2"]]}], ")"}]}]}, { SubscriptBox["\[Gamma]", "1"], RowBox[{ SqrtBox[ RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox[ TagBox["e", Function[{}, E]], SuperscriptBox["S", "2"]]}]], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox[ TagBox["e", Function[{}, E]], SuperscriptBox["S", "2"]]}], ")"}]}]}, { SubscriptBox["\[Gamma]", "2"], RowBox[{ RowBox[{ SuperscriptBox[ TagBox["e", Function[{}, E]], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ TagBox["e", Function[{}, E]], SuperscriptBox["S", "2"]], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox[ TagBox["e", Function[{}, E]], SuperscriptBox["S", "2"]]}], ")"}]}], "+", "3"}], ")"}]}], "-", "6"}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[2.0999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]], TraditionalForm], TraditionalForm, Editable->True], TraditionalForm]], "Output",ExpressionUUID->"98ae86e1-\ 5924-49ea-aee9-e130b9e9bec3"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Raw Moments", "Section",ExpressionUUID->"28c65ce8-4c18-4fad-b3f1-bd66c0823ccb"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}]], "Input",ExpressionUUID->"edde3a3d-\ 4c50-400b-bf9f-d8b619b93e7b"], Cell[BoxData["1"], "Output",ExpressionUUID->"8275777d-a563-49e9-aad9-140d215f9857"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"x", "^", "n"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}]], "Input",ExpressionUUID->"fd1c92cf-\ 8666-4034-af7e-858fa15f60f3"], Cell[BoxData[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"M", " ", "n"}], "+", FractionBox[ RowBox[{ SuperscriptBox["n", "2"], " ", SuperscriptBox["S", "2"]}], "2"]}]]], "Output",ExpressionUUID->"c5ca088a-\ 73c4-4561-83e7-f25097db233c"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Central Moments", "Section",ExpressionUUID->"da9b5bed-97c9-46b7-9c34-8a0966d44517"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["\[Mu]", "1"], ",", SubscriptBox["\[Mu]", "2"], ",", SubscriptBox["\[Mu]", "3"], ",", SubscriptBox["\[Mu]", "4"]}], "}"}], "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], ")"}], "n"]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "4"}], "}"}]}], "]"}]}], ")"}], "//", "ColumnForm"}]], "Input",ExpressionUUID->"a6fcf273-b23f-48d2-b45c-\ 91dcc189ad91"], Cell[BoxData[ InterpretationBox[GridBox[{ {"0"}, { RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"2", " ", "M"}], "+", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}]}, { RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"3", " ", "M"}], "+", FractionBox[ RowBox[{"3", " ", SuperscriptBox["S", "2"]}], "2"]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}], "2"], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}]}, { RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"4", " ", "M"}], "+", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{ SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}]}], ")"}]}]}], ")"}]}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], ColumnForm[{ 0, E^(2 $CellContext`M + $CellContext`S^2) (-1 + E^($CellContext`S^2)), E^(3 $CellContext`M + Rational[3, 2] $CellContext`S^2) (-1 + E^($CellContext`S^2))^2 (2 + E^($CellContext`S^2)), E^(4 $CellContext`M + 2 $CellContext`S^2) (-1 + E^($CellContext`S^2))^2 (-3 + E^(2 $CellContext`S^2) (3 + E^($CellContext`S^2) (2 + E^($CellContext`S^2))))}], Editable->False]], "Output",ExpressionUUID->"766c3b61-ff2a-4f38-9b03-\ c85507b20c1a"] }, Open ]], Cell[CellGroupData[{ Cell["v4.2.1", "Subsubsection",ExpressionUUID->"3e2219f4-2057-416e-94e4-6842fecbd671"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], ")"}], "n"]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}], "//", "Timing"}]], "Input",Expression\ UUID->"af744df2-891e-4f36-90ec-7075b5dd93a3"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"1.9200000000000004`", " ", "Second"}], ",", FractionBox[ RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], "+", "x"}], ")"}], "n"]}], "x"], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S"}]]}], "}"}]], "Output",Expression\ UUID->"90b3a125-b239-418d-baa4-dc98986246d0"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["v5.0.1", "Subsubsection",ExpressionUUID->"dd86d56f-f8b8-4a7b-a7cc-c6042e68a500"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], ")"}], "n"]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}], "//", "Timing"}]], "Input",Expression\ UUID->"1d9d85a1-b69c-49f8-a114-4e9117d5747e"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"219.15`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], "+", "x"}], ")"}], "n"]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID->\ "73e21cc4-b504-4a6b-b34b-85a01d0e96ac"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["v5.1", "Subsubsection",ExpressionUUID->"28e83f79-158e-436d-8c5b-ef46d6386fe3"], Cell[BoxData[ RowBox[{ RowBox[{"Developer`SetSystemOptions", "[", RowBox[{"\"\\"", " ", "->", " ", "True"}], "]"}], ";"}]], "Input",ExpressionUUID->"78fe46b5-d02c-4d13-a089-7f22afc33529"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], ")"}], "n"]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}], "//", "Timing"}]], "Input",Expression\ UUID->"9f9b6e8d-96b2-4aef-a455-d9c7a9a457b7"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"398.83`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], "+", "x"}], ")"}], "n"]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID->\ "1ab857e7-b54c-410b-b9f0-d76c00b5ae23"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["v6.0", "Subsubsection",ExpressionUUID->"a0eb0afb-0b9e-4c2d-a462-fb15da6805c6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], ")"}], "n"]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}], "//", "Timing"}]], "Input",Expression\ UUID->"ab93e12b-4587-4778-8e9b-937f994396d1"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"327.34000000000003`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]]}], "+", "x"}], ")"}], "n"]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"S", ">", "0"}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID->\ "54d5373e-5423-4734-a4de-204a384b1411"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Mean, etc.", "Section",ExpressionUUID->"a68dba77-cbd4-474d-b15b-eaa9c533cd1a"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"4", " ", "M"}], "+", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}], "2"], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", SuperscriptBox["S", "2"]}]]}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"4", " ", SuperscriptBox["S", "2"]}]]}], ")"}], "/", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"2", " ", "M"}], "+", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}], ")"}], "^", "2"}]}]}], "-", "3"}], "]"}]], "Input",ExpressionUUID->"42aa31f4-2273-459b-90e3-\ 721240cd3e08"], Cell[BoxData[ RowBox[{ RowBox[{"-", "6"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", SuperscriptBox["S", "2"]}]]}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"4", " ", SuperscriptBox["S", "2"]}]]}]], "Output",ExpressionUUID->"a88dcb40-e8f0-\ 47be-a855-a3b0d4f0581c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"4", " ", "M"}], "+", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]}]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}], "2"], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", SuperscriptBox["S", "2"]}]]}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"4", " ", SuperscriptBox["S", "2"]}]]}], ")"}], "/", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"2", " ", "M"}], "+", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}], ")"}], "^", "2"}]}]}], "-", "3"}], "]"}]], "Input",ExpressionUUID->"97346a9e-536c-4da9-8bd7-\ 55841b295b7c"], Cell[BoxData[ RowBox[{ RowBox[{"-", "6"}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{ SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}]}], ")"}]}]}]], "Output",Expressi\ onUUID->"de61e292-a516-408d-a3c6-23d04456e378"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"Through", "[", RowBox[{ RowBox[{"{", RowBox[{ "Mean", ",", "Variance", ",", "Skewness", ",", "KurtosisExcess"}], "}"}], "[", RowBox[{"LogNormalDistribution", "[", RowBox[{"M", ",", "S"}], "]"}], "]"}], "]"}], "]"}]], "Input",ExpressionU\ UID->"f5112711-dcd8-448b-a25a-fa8d47b0b3fa"], Cell[BoxData[ RowBox[{"{", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"M", "+", FractionBox[ SuperscriptBox["S", "2"], "2"]}]], ",", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"2", " ", "M"}], "+", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}], ",", RowBox[{ SqrtBox[ RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}]], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}], ",", RowBox[{ RowBox[{"-", "6"}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", SuperscriptBox["S", "2"]}]], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{ SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]], " ", RowBox[{"(", RowBox[{"2", "+", SuperscriptBox["\[ExponentialE]", SuperscriptBox["S", "2"]]}], ")"}]}]}], ")"}]}]}]}], "}"}]], "Output",ExpressionUUID->"233574e3-1917-4c93-984f-fa33fac5f9b5"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Characteristic Function", "Section",ExpressionUUID->"22cc795e-044f-4477-bca8-1c46ab6ca85d"], Cell[CellGroupData[{ Cell["v4.2", "Subsubsection",ExpressionUUID->"d146d309-cf11-49f8-b445-8411bbe30579"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"6146bd6a-41ee-49f8-962f-f1d501db739d"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"7.830000000000001`", " ", "Second"}], ",", RowBox[{ FractionBox["1", RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S"}]], RowBox[{"Integrate", "[", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["M", "2"], "-", RowBox[{"2", " ", "\[ImaginaryI]", " ", SuperscriptBox["S", "2"], " ", "t", " ", "x"}], "+", SuperscriptBox[ RowBox[{"Log", "[", "x", "]"}], "2"]}], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", SuperscriptBox["x", RowBox[{ RowBox[{"-", "1"}], "+", FractionBox["M", SuperscriptBox["S", "2"]]}]]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]}], "}"}], ",", RowBox[{"GenerateConditions", "\[Rule]", "Automatic"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}]}], "}"}]], "Output",ExpressionU\ UID->"f05b2bb1-ad27-47dd-b8b6-864d872fd5c7"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"da9fe8df-090b-4a3c-bec6-c693936376e1"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"2.129999999999999`", " ", "Second"}], ",", FractionBox[ RowBox[{"Integrate", "[", RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], "x"], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S"}]]}], "}"}]], "Output",Expression\ UUID->"4544f088-e646-4f64-ae6c-bf720e37564e"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"FourierTransform", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}]}], ",", "x", ",", "t"}], "]"}], "//", "Timing"}]], "Input",ExpressionUUID->"912ba5f7-d7de-4ba4-9129-\ 99632254b823"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"3.9299999999999997`", " ", "Second"}], ",", FractionBox[ RowBox[{"FourierTransform", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], "x"], ",", "x", ",", "t"}], "]"}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S"}]]}], "}"}]], "Output",Expression\ UUID->"6e76d73a-1047-4fc8-903c-861641cd5e8e"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["v5.0", "Subsubsection",ExpressionUUID->"f3712af2-b99b-4971-bfca-efb29c55f63f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"fd240714-7005-4c72-975d-5b19aae8bc0b"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"18.87207`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID\ ->"a8350714-3d80-49a4-b431-7989ed6a3767"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"0866f301-9527-49ca-b2f6-1945b8b79875"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"305.80078199999997`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID\ ->"6e55b905-b34b-40a5-8129-6a40b9931639"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"FourierTransform", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}]}], ",", "x", ",", "t"}], "]"}], "//", "Timing"}]], "Input",ExpressionUUID->"83db6850-9561-4104-b6ae-\ 7fc313cd325a"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"445.82812399999995`", " ", "Second"}], ",", RowBox[{"FourierTransform", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", "x", ",", "t"}], "]"}]}], "}"}]], "Output",ExpressionUUID->"382f2db3-e301-4d25-\ a68c-33447f55dc97"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["v5.1", "Subsubsection",ExpressionUUID->"2cde65be-09da-4c51-9bce-cd07553cac13"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"3ae35495-bdf6-4f82-b2bb-6f028c63f5d8"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"1200.6000000000001`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID\ ->"88aefdaf-ec72-4010-9ebf-d10b611926ae"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"b6145e3e-c19e-49f8-a9cb-5f14b311a1ff"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"1078.8899999999999`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID\ ->"d75b1607-a840-49e1-9cab-6a1164e9802d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"FourierTransform", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}]}], ",", "x", ",", "t"}], "]"}], "//", "Timing"}]], "Input",ExpressionUUID->"87e2f563-889d-42e9-9f3e-\ 45127308b620"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"305.3000000000002`", " ", "Second"}], ",", RowBox[{"FourierTransform", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", "x", ",", "t"}], "]"}]}], "}"}]], "Output",ExpressionUUID->"57977e55-a45b-4301-\ 8c67-be15811e8b7c"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["v6", "Subsubsection",ExpressionUUID->"9b7c0219-6dd8-4a23-94b8-1cdb9f715ef2"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"ae10ffd7-2e19-4f59-904d-d8e99f597789"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"1041.8399999999997`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID\ ->"7350014f-0a70-4d5d-8432-66f84dc01096"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"Exp", "[", RowBox[{"I", " ", "t", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}], "//", "Timing"}]], "Input",Express\ ionUUID->"ccc2891b-0299-4ecf-83f0-67e9ddf8ca1f"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"934.23`", " ", "Second"}], ",", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "t", " ", "x"}], "-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"M", ">", "0"}], "&&", RowBox[{"S", ">", "0"}]}]}]}], "]"}]}], "}"}]], "Output",ExpressionUUID\ ->"9e7cbe60-becd-4c7c-8e43-53b603ef2b0f"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"FourierTransform", "[", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], RowBox[{"UnitStep", "[", "x", "]"}]}], ",", "x", ",", "t"}], "]"}], "//", "Timing"}]], "Input",ExpressionUUID->"5e7289eb-e657-4a08-9741-\ c0f4e90936b5"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"305.2199999999998`", " ", "Second"}], ",", RowBox[{"FourierTransform", "[", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"M", "-", RowBox[{"Log", "[", "x", "]"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["S", "2"]}]]}]], " ", RowBox[{"UnitStep", "[", "x", "]"}]}], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "S", " ", "x"}]], ",", "x", ",", "t"}], "]"}]}], "}"}]], "Output",ExpressionUUID->"2cc8fc8e-9bd1-4677-\ 87f8-a93659863e5e"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, AutoGeneratedPackage->None, WindowSize->{803, 785}, WindowMargins->{{21, Automatic}, {Automatic, 10}}, FrontEndVersion->"11.1 for Mac OS X x86 (32-bit, 64-bit Kernel) (June 2, \ 2017)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 95, 0, 92, "Title", "ExpressionUUID" -> \ "39cd4b35-a55f-4709-bc06-2803e625c554"], Cell[CellGroupData[{ Cell[700, 26, 83, 0, 44, "Subsection", "ExpressionUUID" -> \ "476dc8de-b39e-481a-9ae6-c7e82b1f70e3"], Cell[786, 28, 111, 3, 49, "Text", "ExpressionUUID" -> \ "3fef5869-824d-4ee0-983c-3320c93efb3a"], Cell[900, 33, 352, 9, 30, "Text", "ExpressionUUID" -> \ "0391f0e6-e6fc-4813-98f5-e5c3dd63044d"], Cell[1255, 44, 372, 10, 30, "Text", "ExpressionUUID" -> \ "7a691740-82cf-4853-ab51-eb596309e6eb"], Cell[1630, 56, 154, 2, 30, "Text", "ExpressionUUID" -> \ "3478df57-8d99-4c52-9a52-d96423a2f325"] }, Open ]], Cell[CellGroupData[{ Cell[1821, 63, 79, 0, 64, "Section", "ExpressionUUID" -> \ "1586a345-0bd2-4902-a70e-c172014c6101"], Cell[CellGroupData[{ Cell[1925, 67, 1092, 27, 117, "Input", "ExpressionUUID" -> \ "f7ef3eed-becb-49b5-a221-8cde03e52026"], Cell[3020, 96, 10236, 422, 156, 8175, 388, "GraphicsData", "PostScript", \ "Graphics", "ExpressionUUID" -> "9f4c6d55-c8a5-4a16-ab20-d34ef607eca0"], Cell[13259, 520, 196, 5, 32, "Output", "ExpressionUUID" -> \ "7a9cd686-163c-4e86-a3d8-6a8c4b37379b"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[13504, 531, 79, 0, 64, "Section", "ExpressionUUID" -> \ "edea9a89-6bac-4ef3-9f52-fb0b34c25c24"], Cell[CellGroupData[{ Cell[13608, 535, 334, 9, 54, "Input", "ExpressionUUID" -> \ "52a1fe5f-46b6-4c70-b6ac-21de9f297f09"], Cell[13945, 546, 4003, 132, 240, "Output", "ExpressionUUID" -> \ "98ae86e1-5924-49ea-aee9-e130b9e9bec3"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[17997, 684, 85, 0, 70, "Section", "ExpressionUUID" -> \ "28c65ce8-4c18-4fad-b3f1-bd66c0823ccb"], Cell[CellGroupData[{ Cell[18107, 688, 656, 20, 70, "Input", "ExpressionUUID" -> \ "edde3a3d-4c50-400b-bf9f-d8b619b93e7b"], Cell[18766, 710, 83, 0, 70, "Output", "ExpressionUUID" -> \ "8275777d-a563-49e9-aad9-140d215f9857"] }, Open ]], Cell[CellGroupData[{ Cell[18886, 715, 713, 22, 70, "Input", "ExpressionUUID" -> \ "fd1c92cf-8666-4034-af7e-858fa15f60f3"], Cell[19602, 739, 270, 8, 70, "Output", "ExpressionUUID" -> \ "c5ca088a-73c4-4561-83e7-f25097db233c"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[19921, 753, 89, 0, 70, "Section", "ExpressionUUID" -> \ "da9b5bed-97c9-46b7-9c34-8a0966d44517"], Cell[CellGroupData[{ Cell[20035, 757, 1422, 42, 70, "Input", "ExpressionUUID" -> \ "a6fcf273-b23f-48d2-b45c-91dcc189ad91"], Cell[21460, 801, 2526, 75, 70, "Output", "ExpressionUUID" -> \ "766c3b61-ff2a-4f38-9b03-c85507b20c1a"] }, Open ]], Cell[CellGroupData[{ Cell[24023, 881, 86, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "3e2219f4-2057-416e-94e4-6842fecbd671"], Cell[CellGroupData[{ Cell[24134, 885, 954, 29, 70, "Input", "ExpressionUUID" -> \ "af744df2-891e-4f36-90ec-7075b5dd93a3"], Cell[25091, 916, 1185, 36, 70, "Output", "ExpressionUUID" -> \ "90b3a125-b239-418d-baa4-dc98986246d0"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[26325, 958, 86, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "dd86d56f-f8b8-4a7b-a7cc-c6042e68a500"], Cell[CellGroupData[{ Cell[26436, 962, 954, 29, 70, "Input", "ExpressionUUID" -> \ "1d9d85a1-b69c-49f8-a114-4e9117d5747e"], Cell[27393, 993, 1083, 33, 70, "Output", "ExpressionUUID" -> \ "73e21cc4-b504-4a6b-b34b-85a01d0e96ac"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[28525, 1032, 84, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "28e83f79-158e-436d-8c5b-ef46d6386fe3"], Cell[28612, 1034, 214, 4, 70, "Input", "ExpressionUUID" -> \ "78fe46b5-d02c-4d13-a089-7f22afc33529"], Cell[CellGroupData[{ Cell[28851, 1042, 954, 29, 70, "Input", "ExpressionUUID" -> \ "9f9b6e8d-96b2-4aef-a455-d9c7a9a457b7"], Cell[29808, 1073, 1083, 33, 70, "Output", "ExpressionUUID" -> \ "1ab857e7-b54c-410b-b9f0-d76c00b5ae23"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[30940, 1112, 84, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "a0eb0afb-0b9e-4c2d-a462-fb15da6805c6"], Cell[CellGroupData[{ Cell[31049, 1116, 954, 29, 70, "Input", "ExpressionUUID" -> \ "ab93e12b-4587-4778-8e9b-937f994396d1"], Cell[32006, 1147, 1095, 33, 70, "Output", "ExpressionUUID" -> \ "54d5373e-5423-4734-a4de-204a384b1411"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[33162, 1187, 84, 0, 70, "Section", "ExpressionUUID" -> \ "a68dba77-cbd4-474d-b15b-eaa9c533cd1a"], Cell[CellGroupData[{ Cell[33271, 1191, 1377, 43, 70, "Input", "ExpressionUUID" -> \ "42aa31f4-2273-459b-90e3-721240cd3e08"], Cell[34651, 1236, 461, 14, 70, "Output", "ExpressionUUID" -> \ "a88dcb40-e8f0-47be-a855-a3b0d4f0581c"] }, Open ]], Cell[CellGroupData[{ Cell[35149, 1255, 1381, 43, 70, "Input", "ExpressionUUID" -> \ "97346a9e-536c-4da9-8bd7-55841b295b7c"], Cell[36533, 1300, 515, 16, 70, "Output", "ExpressionUUID" -> \ "de61e292-a516-408d-a3c6-23d04456e378"] }, Open ]], Cell[CellGroupData[{ Cell[37085, 1321, 373, 10, 70, "Input", "ExpressionUUID" -> \ "f5112711-dcd8-448b-a25a-fa8d47b0b3fa"], Cell[37461, 1333, 1295, 42, 70, "Output", "ExpressionUUID" -> \ "233574e3-1917-4c93-984f-fa33fac5f9b5"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[38805, 1381, 97, 0, 70, "Section", "ExpressionUUID" -> \ "22cc795e-044f-4477-bca8-1c46ab6ca85d"], Cell[CellGroupData[{ Cell[38927, 1385, 84, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "d146d309-cf11-49f8-b445-8411bbe30579"], Cell[CellGroupData[{ Cell[39036, 1389, 939, 28, 70, "Input", "ExpressionUUID" -> \ "6146bd6a-41ee-49f8-962f-f1d501db739d"], Cell[39978, 1419, 1281, 37, 70, "Output", "ExpressionUUID" -> \ "f05b2bb1-ad27-47dd-b8b6-864d872fd5c7"] }, Open ]], Cell[CellGroupData[{ Cell[41296, 1461, 864, 26, 70, "Input", "ExpressionUUID" -> \ "da9fe8df-090b-4a3c-bec6-c693936376e1"], Cell[42163, 1489, 985, 29, 70, "Output", "ExpressionUUID" -> \ "4544f088-e646-4f64-ae6c-bf720e37564e"] }, Open ]], Cell[CellGroupData[{ Cell[43185, 1523, 649, 20, 70, "Input", "ExpressionUUID" -> \ "912ba5f7-d7de-4ba4-9129-99632254b823"], Cell[43837, 1545, 757, 23, 70, "Output", "ExpressionUUID" -> \ "6e76d73a-1047-4fc8-903c-861641cd5e8e"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[44643, 1574, 84, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "f3712af2-b99b-4971-bfca-efb29c55f63f"], Cell[CellGroupData[{ Cell[44752, 1578, 939, 28, 70, "Input", "ExpressionUUID" -> \ "fd240714-7005-4c72-975d-5b19aae8bc0b"], Cell[45694, 1608, 1003, 29, 70, "Output", "ExpressionUUID" -> \ "a8350714-3d80-49a4-b431-7989ed6a3767"] }, Open ]], Cell[CellGroupData[{ Cell[46734, 1642, 864, 26, 70, "Input", "ExpressionUUID" -> \ "0866f301-9527-49ca-b2f6-1945b8b79875"], Cell[47601, 1670, 903, 26, 70, "Output", "ExpressionUUID" -> \ "6e55b905-b34b-40a5-8129-6a40b9931639"] }, Open ]], Cell[CellGroupData[{ Cell[48541, 1701, 649, 20, 70, "Input", "ExpressionUUID" -> \ "83db6850-9561-4104-b6ae-7fc313cd325a"], Cell[49193, 1723, 737, 22, 70, "Output", "ExpressionUUID" -> \ "382f2db3-e301-4d25-a68c-33447f55dc97"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[49979, 1751, 84, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "2cde65be-09da-4c51-9bce-cd07553cac13"], Cell[CellGroupData[{ Cell[50088, 1755, 939, 28, 70, "Input", "ExpressionUUID" -> \ "3ae35495-bdf6-4f82-b2bb-6f028c63f5d8"], Cell[51030, 1785, 1013, 29, 70, "Output", "ExpressionUUID" -> \ "88aefdaf-ec72-4010-9ebf-d10b611926ae"] }, Open ]], Cell[CellGroupData[{ Cell[52080, 1819, 864, 26, 70, "Input", "ExpressionUUID" -> \ "b6145e3e-c19e-49f8-a9cb-5f14b311a1ff"], Cell[52947, 1847, 903, 26, 70, "Output", "ExpressionUUID" -> \ "d75b1607-a840-49e1-9cab-6a1164e9802d"] }, Open ]], Cell[CellGroupData[{ Cell[53887, 1878, 649, 20, 70, "Input", "ExpressionUUID" -> \ "87e2f563-889d-42e9-9f3e-45127308b620"], Cell[54539, 1900, 736, 22, 70, "Output", "ExpressionUUID" -> \ "57977e55-a45b-4301-8c67-be15811e8b7c"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[55324, 1928, 82, 0, 70, "Subsubsection", "ExpressionUUID" -> \ "9b7c0219-6dd8-4a23-94b8-1cdb9f715ef2"], Cell[CellGroupData[{ Cell[55431, 1932, 939, 28, 70, "Input", "ExpressionUUID" -> \ "ae10ffd7-2e19-4f59-904d-d8e99f597789"], Cell[56373, 1962, 1013, 29, 70, "Output", "ExpressionUUID" -> \ "7350014f-0a70-4d5d-8432-66f84dc01096"] }, Open ]], Cell[CellGroupData[{ Cell[57423, 1996, 864, 26, 70, "Input", "ExpressionUUID" -> \ "ccc2891b-0299-4ecf-83f0-67e9ddf8ca1f"], Cell[58290, 2024, 891, 26, 70, "Output", "ExpressionUUID" -> \ "9e7cbe60-becd-4c7c-8e43-53b603ef2b0f"] }, Open ]], Cell[CellGroupData[{ Cell[59218, 2055, 649, 20, 70, "Input", "ExpressionUUID" -> \ "5e7289eb-e657-4a08-9741-c0f4e90936b5"], Cell[59870, 2077, 736, 22, 70, "Output", "ExpressionUUID" -> \ "2cc8fc8e-9bd1-4677-87f8-a93659863e5e"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)