summaryrefslogtreecommitdiff
path: root/3d_ising_analytic_work.nb
diff options
context:
space:
mode:
authorJaron Kent-Dobias <jaron@kent-dobias.com>2021-08-17 19:10:01 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2021-08-17 19:10:01 +0200
commitd843c9d2d5c25ecc10760b1fc834348b1290921b (patch)
treebaa27c4cb0fa311872d3ac515723f27993daabb3 /3d_ising_analytic_work.nb
parentd0796ec87f2bc0f57c2ba76497918f4c570dbd46 (diff)
downloadmma-d843c9d2d5c25ecc10760b1fc834348b1290921b.tar.gz
mma-d843c9d2d5c25ecc10760b1fc834348b1290921b.tar.bz2
mma-d843c9d2d5c25ecc10760b1fc834348b1290921b.zip
Lots of work.
Diffstat (limited to '3d_ising_analytic_work.nb')
-rw-r--r--3d_ising_analytic_work.nb1168
1 files changed, 1168 insertions, 0 deletions
diff --git a/3d_ising_analytic_work.nb b/3d_ising_analytic_work.nb
new file mode 100644
index 0000000..a163ee6
--- /dev/null
+++ b/3d_ising_analytic_work.nb
@@ -0,0 +1,1168 @@
+(* Content-type: application/vnd.wolfram.mathematica *)
+
+(*** Wolfram Notebook File ***)
+(* http://www.wolfram.com/nb *)
+
+(* CreatedBy='Mathematica 12.3' *)
+
+(*CacheID: 234*)
+(* Internal cache information:
+NotebookFileLineBreakTest
+NotebookFileLineBreakTest
+NotebookDataPosition[ 158, 7]
+NotebookDataLength[ 42611, 1158]
+NotebookOptionsPosition[ 39591, 1098]
+NotebookOutlinePosition[ 39990, 1114]
+CellTagsIndexPosition[ 39947, 1111]
+WindowFrame->Normal*)
+
+(* Beginning of Notebook Content *)
+Notebook[{
+
+Cell[CellGroupData[{
+Cell[BoxData[
+ RowBox[{"Integrate", "[",
+ RowBox[{
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["x",
+ RowBox[{
+ RowBox[{"-", "7"}], "/", "3"}]],
+ RowBox[{"Exp", "[",
+ RowBox[{
+ RowBox[{"-", "1"}], "/",
+ SuperscriptBox["x", "2"]}], "]"}]}],
+ RowBox[{"(",
+ RowBox[{"x", "+", "y"}], ")"}]], ",",
+ RowBox[{"{",
+ RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",",
+ RowBox[{"Assumptions", "->",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"x", ">", "0"}], ",",
+ RowBox[{"y", ">", "0"}], ",",
+ RowBox[{"yc", ">", "0"}]}], "}"}]}]}], "]"}]], "Input",
+ CellChangeTimes->{3.8371541989870367`*^9},
+ CellLabel->"In[48]:=",ExpressionUUID->"fababbb8-381a-4687-8bb1-07b91e8a5e30"],
+
+Cell[BoxData[
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["y", "2"]]}]], " ",
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{
+ RowBox[{"-",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["7", "6"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["y", "2"]]}]}], "]"}]}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "+",
+ RowBox[{"4", " ", "y", " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["y", "2"]]}]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}]}], ")"}]}],
+ RowBox[{"12", " ",
+ SuperscriptBox["y", "2"]}]]], "Output",
+ CellChangeTimes->{3.837154204903788*^9},
+ CellLabel->"Out[48]=",ExpressionUUID->"18ce2878-724a-4922-9798-030bcc1e0186"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"i1", "=",
+ RowBox[{
+ FractionBox[
+ SuperscriptBox["\[Theta]", "2"], "\[Pi]"],
+ RowBox[{"Integrate", "[",
+ RowBox[{
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["x",
+ RowBox[{
+ RowBox[{"-", "7"}], "/", "3"}]],
+ RowBox[{"Exp", "[",
+ RowBox[{
+ RowBox[{"-", "1"}], "/",
+ SuperscriptBox["x", "2"]}], "]"}]}],
+ RowBox[{
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"x", "+", "yc"}], ")"}], "2"],
+ RowBox[{"(",
+ RowBox[{"x", "+", "y"}], ")"}]}]], ",",
+ RowBox[{"{",
+ RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",",
+ RowBox[{"Assumptions", "->",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"x", ">", "0"}], ",",
+ RowBox[{"y", ">", "0"}], ",",
+ RowBox[{"yc", ">", "0"}]}], "}"}]}]}], "]"}]}]}]], "Input",
+ CellChangeTimes->{{3.837152286872525*^9, 3.8371524292104893`*^9}, {
+ 3.837152588581934*^9, 3.8371525930135*^9}, {3.8371526561032867`*^9,
+ 3.8371526576627903`*^9}, {3.837152833506393*^9, 3.837152859458221*^9}, {
+ 3.8371530145983763`*^9, 3.83715307324615*^9}, {3.8371531339360533`*^9,
+ 3.837153158839828*^9}, {3.83715327430684*^9, 3.837153280810217*^9}, {
+ 3.837153503855143*^9, 3.837153504046052*^9}},
+ CellLabel->"In[37]:=",ExpressionUUID->"bf6da1af-2ff5-46b6-a7be-85f3bdd44d14"],
+
+Cell[BoxData[
+ RowBox[{
+ FractionBox["1",
+ RowBox[{"36", " ", "\[Pi]", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"y", "-", "yc"}], ")"}], "2"]}]],
+ RowBox[{
+ SuperscriptBox["\[Theta]", "2"], " ",
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{
+ FractionBox["1",
+ SuperscriptBox["y",
+ RowBox[{"7", "/", "3"}]]],
+ RowBox[{"3", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["y", "2"]]}]], " ",
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"8", " ",
+ SqrtBox["3"], " ", "\[Pi]"}], "+",
+ RowBox[{
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["1", "6"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "-",
+ RowBox[{"4", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["2", "3"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "+",
+ RowBox[{"4", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["2", "3"]}], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["y", "2"]]}]}], "]"}]}], "-",
+ RowBox[{
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["1", "6"]}], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["y", "2"]]}]}], "]"}]}]}], ")"}]}]}], "-",
+ RowBox[{
+ FractionBox["1",
+ SuperscriptBox["yc",
+ RowBox[{"7", "/", "3"}]]],
+ RowBox[{"3", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]], " ",
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"8", " ",
+ SqrtBox["3"], " ", "\[Pi]"}], "+",
+ RowBox[{
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["1", "6"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "-",
+ RowBox[{"4", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["2", "3"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "+",
+ RowBox[{"4", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["2", "3"]}], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]}], "]"}]}], "-",
+ RowBox[{
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["1", "6"]}], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]}], "]"}]}]}], ")"}]}]}], "+",
+ RowBox[{
+ FractionBox["1",
+ SuperscriptBox["yc",
+ RowBox[{"16", "/", "3"}]]],
+ RowBox[{
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]], " ",
+ RowBox[{"(",
+ RowBox[{"y", "-", "yc"}], ")"}], " ",
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{
+ RowBox[{"-", "48"}], " ",
+ SqrtBox["3"], " ", "\[Pi]"}], "+",
+ RowBox[{"56", " ",
+ SqrtBox["3"], " ", "\[Pi]", " ",
+ SuperscriptBox["yc", "2"]}], "+",
+ RowBox[{"6", " ",
+ SuperscriptBox["\[ExponentialE]",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]], " ",
+ SuperscriptBox["yc",
+ RowBox[{"7", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "-",
+ RowBox[{"6", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["1", "6"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "+",
+ RowBox[{"7", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ SuperscriptBox["yc", "2"], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["1", "6"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "-",
+ RowBox[{"36", " ",
+ SuperscriptBox["\[ExponentialE]",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]], " ",
+ SuperscriptBox["yc",
+ RowBox[{"4", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "-",
+ RowBox[{"12", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ SuperscriptBox["yc", "2"], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{"-",
+ FractionBox["2", "3"]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "-",
+ RowBox[{"36", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "+",
+ RowBox[{"24", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ SuperscriptBox["yc", "2"], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "+",
+ RowBox[{"12", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ SuperscriptBox["yc", "2"], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["2", "3"]}], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]}], "]"}]}], "+",
+ RowBox[{
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"1", "/", "6"}]], " ",
+ RowBox[{"(",
+ RowBox[{"6", "-",
+ RowBox[{"7", " ",
+ SuperscriptBox["yc", "2"]}]}], ")"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["1", "6"]}], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]}], "]"}]}], "+",
+ RowBox[{"36", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ FractionBox["1", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]}], "]"}]}], "-",
+ RowBox[{"24", " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"-", "1"}], ")"}],
+ RowBox[{"2", "/", "3"}]], " ",
+ SuperscriptBox["yc", "2"], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ FractionBox["1", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox["yc", "2"]]}]}], "]"}]}]}], ")"}]}]}]}],
+ ")"}]}]}]], "Output",
+ CellChangeTimes->{
+ 3.83715241847911*^9, 3.837152473882375*^9, 3.837152648741281*^9,
+ 3.837152714070775*^9, 3.8371529158052588`*^9, 3.8371530773164387`*^9, {
+ 3.837153147250849*^9, 3.8371531813231363`*^9}, {3.8371532786171103`*^9,
+ 3.83715328927947*^9}, 3.837153512224599*^9},
+ CellLabel->"Out[37]=",ExpressionUUID->"efae8b97-cac0-4dfb-b6a6-403456514c4f"]
+}, Open ]],
+
+Cell[BoxData[
+ RowBox[{"i2", "=",
+ RowBox[{"FullSimplify", "[",
+ RowBox[{
+ RowBox[{
+ RowBox[{"(",
+ RowBox[{"i1", "/.",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"y", "->",
+ RowBox[{"b",
+ RowBox[{"(",
+ RowBox[{"\[Theta]c", "-", "\[Theta]"}], ")"}]}]}], ",",
+ RowBox[{"yc", "->",
+ RowBox[{"b", " ", "\[Theta]c"}]}]}], "}"}]}], ")"}], "+",
+ RowBox[{"(",
+ RowBox[{"i1", "/.",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"y", "->",
+ RowBox[{"b",
+ RowBox[{"(",
+ RowBox[{"\[Theta]c", "+", "\[Theta]"}], ")"}]}]}], ",",
+ RowBox[{"yc", "->",
+ RowBox[{"b", " ", "\[Theta]c"}]}]}], "}"}]}], ")"}]}], ",",
+ RowBox[{"Assumptions", "->",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"b", ">", "0"}], ",",
+ RowBox[{"\[Theta]c", ">", "0"}], ",",
+ RowBox[{"\[Theta]", "<", "\[Theta]c"}], ",",
+ RowBox[{"\[Theta]", ">",
+ RowBox[{"-", "\[Theta]c"}]}]}], "}"}]}]}], "]"}]}]], "Input",
+ CellChangeTimes->{{3.837153261498691*^9, 3.8371533873000507`*^9}, {
+ 3.837153547543831*^9, 3.837153576103598*^9}, 3.8371549111209583`*^9},
+ CellLabel->"In[52]:=",ExpressionUUID->"c8a865bb-cad1-450c-8187-a809f109c3c8"],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"iSing", "=",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{"12", " ", "\[Pi]"}]]}],
+ RowBox[{"(",
+ RowBox[{
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["7", "6"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]}],
+ "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}],
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]], "+",
+ FractionBox[
+ RowBox[{"4", " ", "b", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]}],
+ "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}],
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}]]}], ")"}]}]}]], "Input",
+ CellChangeTimes->{{3.837159003786721*^9, 3.837159023124399*^9}},
+ CellLabel->"In[56]:=",ExpressionUUID->"bd0a8b22-65f5-4df7-b577-0bf969c52194"],
+
+Cell[BoxData[
+ RowBox[{"-",
+ FractionBox[
+ RowBox[{
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["7", "6"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]}],
+ "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}],
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]], "+",
+ FractionBox[
+ RowBox[{"4", " ", "b", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]}],
+ "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}],
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}]]}],
+ RowBox[{"12", " ", "\[Pi]"}]]}]], "Output",
+ CellChangeTimes->{{3.837159019621305*^9, 3.837159023641695*^9}},
+ CellLabel->"Out[56]=",ExpressionUUID->"3576d1fd-5496-4abf-b6de-d73b330829ac"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{
+ RowBox[{"SeriesCoefficient", "[",
+ RowBox[{"iSing", ",",
+ RowBox[{"{",
+ RowBox[{"\[Theta]", ",", "\[Theta]c", ",", "1"}], "}"}], ",",
+ RowBox[{"Assumptions", "->",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"\[Theta]c", ">", "0"}], ",",
+ RowBox[{"\[Theta]", "<", "\[Theta]c"}], ",",
+ RowBox[{"b", ">", "0"}]}], "}"}]}]}], "]"}], "//", "N"}]], "Input",
+ CellChangeTimes->{{3.837159089333461*^9, 3.8371591098128653`*^9}, {
+ 3.837159228863208*^9, 3.8371592353512363`*^9}, {3.837159280672409*^9,
+ 3.837159285872224*^9}, {3.837159400090641*^9, 3.83715940032231*^9}, {
+ 3.8371594403473797`*^9, 3.837159450811274*^9}},
+ CellLabel->"In[81]:=",ExpressionUUID->"430dc047-855a-4d97-9e15-6a1b8ed47326"],
+
+Cell[BoxData[
+ RowBox[{"0.14367637572608224`", " ",
+ SuperscriptBox["b", "3"]}]], "Output",
+ CellChangeTimes->{
+ 3.837159110237748*^9, {3.837159229380753*^9, 3.837159235660652*^9}, {
+ 3.837159281262404*^9, 3.837159286279546*^9}, 3.8371594008606462`*^9, {
+ 3.837159440963653*^9, 3.837159451210906*^9}},
+ CellLabel->"Out[81]=",ExpressionUUID->"4c7ac351-a724-4419-9c0c-ec0428c101c7"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"t1", "=",
+ RowBox[{
+ FractionBox["1", "\[Pi]"],
+ RowBox[{"Integrate", "[",
+ RowBox[{
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["x",
+ RowBox[{
+ RowBox[{"-", "7"}], "/", "3"}]],
+ RowBox[{"Exp", "[",
+ RowBox[{
+ RowBox[{"-", "1"}], "/",
+ SuperscriptBox["x", "2"]}], "]"}]}],
+ SuperscriptBox["x",
+ RowBox[{"n", "+", "1"}]]], ",",
+ RowBox[{"{",
+ RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}], ",",
+ RowBox[{"Assumptions", "->",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"x", ">", "0"}], ",",
+ RowBox[{"y", ">", "0"}], ",",
+ RowBox[{"yc", ">", "0"}], ",",
+ RowBox[{"n", ">", "0"}], ",",
+ RowBox[{"n", "\[Element]", "Integers"}]}], "}"}]}]}],
+ "]"}]}]}]], "Input",
+ CellChangeTimes->{{3.837159052820043*^9, 3.837159080308568*^9}, {
+ 3.837159125069777*^9, 3.837159126237418*^9}, {3.837159159774912*^9,
+ 3.837159167254078*^9}, {3.837159215647285*^9, 3.8371592157911263`*^9}, {
+ 3.8371592706725616`*^9, 3.8371592715678377`*^9}},
+ CellLabel->"In[71]:=",ExpressionUUID->"608d5852-49d9-45d7-9763-cb10da65d830"],
+
+Cell[BoxData[
+ FractionBox[
+ RowBox[{"Gamma", "[",
+ RowBox[{
+ FractionBox["7", "6"], "+",
+ FractionBox["n", "2"]}], "]"}],
+ RowBox[{"2", " ", "\[Pi]"}]]], "Output",
+ CellChangeTimes->{{3.837159072418518*^9, 3.83715908203614*^9},
+ 3.837159127950366*^9, 3.837159168943486*^9, 3.837159217563138*^9,
+ 3.837159273575577*^9},
+ CellLabel->"Out[71]=",ExpressionUUID->"6e748a34-f946-4d43-8801-0833c28048e5"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{
+ RowBox[{"t1", "/.",
+ RowBox[{"n", "->", "0"}]}], "//", "N"}]], "Input",
+ CellChangeTimes->{{3.837159127206106*^9, 3.837159130805312*^9}, {
+ 3.837159222143116*^9, 3.837159238231101*^9}, {3.8371592776402197`*^9,
+ 3.837159288088141*^9}, {3.837159404642499*^9, 3.8371594052183743`*^9}},
+ CellLabel->"In[78]:=",ExpressionUUID->"5fe4ff16-26ea-47b3-a961-9c776acf99bb"],
+
+Cell[BoxData["0.14765111774913997`"], "Output",
+ CellChangeTimes->{
+ 3.837159131124867*^9, 3.8371591696139517`*^9, {3.837159218839643*^9,
+ 3.8371592385873203`*^9}, {3.837159274439986*^9, 3.837159288334805*^9},
+ 3.837159405552266*^9},
+ CellLabel->"Out[78]=",ExpressionUUID->"cffc2fe3-b3b9-420d-ae4c-5559b2e114bb"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{
+ RowBox[{
+ FractionBox[
+ RowBox[{"4", " ", "b", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}], ")"}], "2"]}]]}]}], "]"}],
+ " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}],
+ RowBox[{"\[Theta]", "-", "\[Theta]c"}]], "+",
+ RowBox[{
+ RowBox[{"(",
+ RowBox[{"-",
+ FractionBox[
+ RowBox[{"4", " ", "b", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "+", "\[Theta]c"}], ")"}], "2"]}]]}]], " ",
+
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"\[Theta]", "+", "\[Theta]c"}], ")"}], "2"]}]]}]}],
+ "]"}]}],
+ RowBox[{"\[Theta]", "+", "\[Theta]c"}]]}], ")"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}]}], "/.",
+ RowBox[{"\[Theta]", "->", "0"}]}]], "Input",
+ CellChangeTimes->{{3.837156673000149*^9, 3.837156678457199*^9}},
+ CellLabel->"In[54]:=",ExpressionUUID->"c1119876-e977-4ca0-8963-ee017dea27fd"],
+
+Cell[BoxData[
+ RowBox[{"-",
+ FractionBox[
+ RowBox[{"8", " ", "b", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox["\[Theta]c", "2"]}]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ RowBox[{
+ SuperscriptBox["b", "2"], " ",
+ SuperscriptBox["\[Theta]c", "2"]}]]}]}], "]"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}], "\[Theta]c"]}]], "Output",
+ CellChangeTimes->{{3.837156674332121*^9, 3.837156679177827*^9}},
+ CellLabel->"Out[54]=",ExpressionUUID->"af9d10e0-c1b2-44e5-b2ed-d286376e73b8"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"i2", "/.",
+ RowBox[{"\[Theta]", "->", "0"}]}]], "Input",
+ CellChangeTimes->{{3.837153471782505*^9, 3.837153475437471*^9}},
+ CellLabel->"In[39]:=",ExpressionUUID->"54955d9a-a1b8-49ea-8b22-6f8c2c06fec5"],
+
+Cell[BoxData["0"], "Output",
+ CellChangeTimes->{3.837153475703751*^9, 3.837153517514927*^9},
+ CellLabel->"Out[39]=",ExpressionUUID->"06308438-871f-452c-aef0-07985232fef7"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"i3", "=",
+ RowBox[{"Simplify", "[",
+ RowBox[{"i2", "/.",
+ RowBox[{"{",
+ RowBox[{
+ RowBox[{"b", "->", "1"}], ",",
+ RowBox[{"\[Theta]c", "->", "1"}]}], "}"}]}], "]"}]}]], "Input",
+ CellChangeTimes->{{3.8371541718671207`*^9, 3.837154174051682*^9}},
+ CellLabel->"In[46]:=",ExpressionUUID->"3b3622a7-fadc-4e42-a347-8de6c9595dd3"],
+
+Cell[BoxData[
+ RowBox[{"-",
+ RowBox[{
+ FractionBox["1",
+ RowBox[{"12", " ", "\[Pi]"}]],
+ RowBox[{"(",
+ RowBox[{
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"-", "1"}], "+", "\[Theta]"}], ")"}], "2"]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["7", "6"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"-", "1"}], "+", "\[Theta]"}], ")"}], "2"]]}]}], "]"}],
+ " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}],
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"-", "1"}], "+", "\[Theta]"}], ")"}], "2"]], "+",
+ RowBox[{
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"-",
+ FractionBox[
+ RowBox[{"2", " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["7", "6"], ",",
+ RowBox[{"-", "1"}]}], "]"}]}], "\[ExponentialE]"]}], "+",
+ FractionBox[
+ RowBox[{
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"1", "+", "\[Theta]"}], ")"}], "2"]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["7", "6"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"1", "+", "\[Theta]"}], ")"}], "2"]]}]}], "]"}]}],
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"1", "+", "\[Theta]"}], ")"}], "2"]]}], ")"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["1", "6"], "]"}]}], "+",
+ FractionBox[
+ RowBox[{"4", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"-", "1"}], "+", "\[Theta]"}], ")"}], "2"]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{
+ RowBox[{"-", "1"}], "+", "\[Theta]"}], ")"}], "2"]]}]}], "]"}],
+ " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}],
+ RowBox[{
+ RowBox[{"-", "1"}], "+", "\[Theta]"}]], "+",
+ RowBox[{
+ RowBox[{"(",
+ RowBox[{
+ FractionBox[
+ RowBox[{"8", " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-", "1"}]}], "]"}]}], "\[ExponentialE]"], "-",
+ FractionBox[
+ RowBox[{"4", " ",
+ SuperscriptBox["\[ExponentialE]",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"1", "+", "\[Theta]"}], ")"}], "2"]]}]], " ",
+ RowBox[{"ExpIntegralE", "[",
+ RowBox[{
+ FractionBox["5", "3"], ",",
+ RowBox[{"-",
+ FractionBox["1",
+ SuperscriptBox[
+ RowBox[{"(",
+ RowBox[{"1", "+", "\[Theta]"}], ")"}], "2"]]}]}], "]"}]}],
+ RowBox[{"1", "+", "\[Theta]"}]]}], ")"}], " ",
+ RowBox[{"Gamma", "[",
+ FractionBox["2", "3"], "]"}]}]}], ")"}]}]}]], "Output",
+ CellChangeTimes->{3.8371541745414057`*^9},
+ CellLabel->"Out[46]=",ExpressionUUID->"caca42d0-5916-43b6-84ea-cc91bc115d17"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"Plot", "[",
+ RowBox[{"i3", ",",
+ RowBox[{"{",
+ RowBox[{"\[Theta]", ",",
+ RowBox[{"-", "1.2"}], ",", "1.2"}], "}"}], ",",
+ RowBox[{"PlotRange", "->", "All"}], ",",
+ RowBox[{"WorkingPrecision", "\[Rule]", "20"}]}], "]"}]], "Input",
+ CellChangeTimes->{{3.8371534184294147`*^9, 3.837153465381342*^9}, {
+ 3.8371538031322393`*^9, 3.8371538039795713`*^9}, {3.837154155242539*^9,
+ 3.8371541836983547`*^9}, {3.837154249916608*^9, 3.837154260835929*^9}},
+ CellLabel->"In[50]:=",ExpressionUUID->"bdc138d2-dd1c-49d9-b891-f23d7c6a21a1"],
+
+Cell[BoxData[
+ GraphicsBox[{{{}, {},
+ TagBox[
+ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
+ 1.], LineBox[CompressedData["
+1:eJwBsQFO/iFib1JlAgAAABoAAAACAAAAaHXmkTf68r9xFzYS7cGzPwE+EGPq
++fK/KsnCecvAsz8oFLKO5vbyvxbpET+AtbM/T+pTuuLz8r+ZTc+TPaqzP3bA
+9eXe8PK/IZNObAOfsz/FbDk91+ryv3OiaHqoiLM/7ELbaNPn8r8z6zCZh32z
+PxQZfZTP5PK/zFEVDm9ysz9ixcDrx97yv+hAqs1WXLM//h1ImrjS8r/WfMBx
+iDCzPyX06cW0z/K/D8JvDqklsz9MyovxsMzyv81nqarRGrM/mnbPSKnG8r9b
+AjW4OgWzPzfPVveZuvK/vTzkCWvasj9wgGVUe6LyvzxQ5d87hrI/4eKCDj5y
+8r9jRgUuXeOxP+4mFE75bvK/xfNMTZXYsT/6aqWNtGvyv3oqu3jVzbE/FPPH
+DCtl8r9GwSTLbbixP0gDDQsYWPK/AaR4RP2NsT+vI5cH8j3yv+h88P2OOrE/
+fWSrAKYJ8r8gmrMQPZmwPxnm0/INofG/qJL99evVrj95ptylut3wv0k0rqP3
+6ao/bg2pNjwe8L83VmndGJSnPw5HTHsiA/C/3y4hdDUkpz9INuaO
+ "]],
+ LineBox[CompressedData["
+1:eJw1l2k01XvcxUkpGZIoFGWWcEyJ6P9FMpXMbmTMVGYu4RSFrplrCsmQ4WQo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+ "]], LineBox[CompressedData["
+1:eJwVj30s1HEAhy+vvSwrtqgtRUkxjqQXL32iKw1/SJj3RM0ppeby0q3GGkUl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+ "]]},
+ Annotation[#, "Charting`Private`Tag$591217#1"]& ], {}}, {}},
+ AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
+ Axes->{True, True},
+ AxesLabel->{None, None},
+ AxesOrigin->{0, 0},
+ DisplayFunction->Identity,
+ Frame->{{False, False}, {False, False}},
+ FrameLabel->{{None, None}, {None, None}},
+ FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
+ GridLines->{None, None},
+ GridLinesStyle->Directive[
+ GrayLevel[0.5, 0.4]],
+ ImagePadding->All,
+ Method->{
+ "DefaultBoundaryStyle" -> Automatic,
+ "DefaultGraphicsInteraction" -> {
+ "Version" -> 1.2, "TrackMousePosition" -> {True, False},
+ "Effects" -> {
+ "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2},
+ "Droplines" -> {
+ "freeformCursorMode" -> True,
+ "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" ->
+ AbsolutePointSize[6], "ScalingFunctions" -> None,
+ "CoordinatesToolOptions" -> {"DisplayFunction" -> ({
+ (Identity[#]& )[
+ Part[#, 1]],
+ (Identity[#]& )[
+ Part[#, 2]]}& ), "CopiedValueFunction" -> ({
+ (Identity[#]& )[
+ Part[#, 1]],
+ (Identity[#]& )[
+ Part[#, 2]]}& )}},
+ PlotRange->{All, All},
+ PlotRangeClipping->True,
+ PlotRangePadding->{{
+ Scaled[0.02],
+ Scaled[0.02]}, {
+ Scaled[0.05],
+ Scaled[0.05]}},
+ Ticks->{Automatic, Automatic}]], "Output",
+ CellChangeTimes->{{3.8371534327807703`*^9, 3.8371534661081553`*^9},
+ 3.8371535186967163`*^9, {3.837153795719611*^9, 3.837153804720337*^9}, {
+ 3.837154159258175*^9, 3.8371541842464027`*^9}, {3.837154252931675*^9,
+ 3.8371542615178967`*^9}},
+ CellLabel->"Out[50]=",ExpressionUUID->"7dab72fd-cd30-44be-b73a-d23ec16b745a"]
+}, Open ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"FullSimplify", "[", "i1", "]"}]], "Input",
+ CellChangeTimes->{{3.837153188209074*^9, 3.837153192208252*^9}},
+ CellLabel->"In[27]:=",ExpressionUUID->"acd277d9-1867-4c72-bd06-edff0291a14e"],
+
+Cell[BoxData["$Aborted"], "Output",
+ CellChangeTimes->{3.837153204428977*^9},
+ CellLabel->"Out[27]=",ExpressionUUID->"a786f1b5-8e89-47a1-ba0e-8efd965ebaa9"]
+}, Open ]],
+
+Cell[BoxData[
+ RowBox[{"i1", "/.",
+ RowBox[{"y", "->"}]}]], "Input",
+ CellChangeTimes->{{3.837153084295353*^9,
+ 3.837153086518475*^9}},ExpressionUUID->"b3b0287b-1e22-4b15-a4f5-\
+10695aa5f35c"],
+
+Cell[BoxData[" "], "Input",
+ CellChangeTimes->{
+ 3.837152869906995*^9},ExpressionUUID->"072b5925-5f74-49be-9a73-\
+34916c763d7c"]
+},
+WindowSize->{636., 350.25},
+WindowMargins->{{Automatic, 2.25}, {357.75, Automatic}},
+FrontEndVersion->"12.3 for Linux x86 (64-bit) (May 11, 2021)",
+StyleDefinitions->"Default.nb",
+ExpressionUUID->"72b00315-9f0b-4e16-a7c6-aa74d14cd275"
+]
+(* End of Notebook Content *)
+
+(* Internal cache information *)
+(*CellTagsOutline
+CellTagsIndex->{}
+*)
+(*CellTagsIndex
+CellTagsIndex->{}
+*)
+(*NotebookFileOutline
+Notebook[{
+Cell[CellGroupData[{
+Cell[580, 22, 747, 23, 41, "Input",ExpressionUUID->"fababbb8-381a-4687-8bb1-07b91e8a5e30"],
+Cell[1330, 47, 989, 31, 61, "Output",ExpressionUUID->"18ce2878-724a-4922-9798-030bcc1e0186"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[2356, 83, 1367, 36, 42, "Input",ExpressionUUID->"bf6da1af-2ff5-46b6-a7be-85f3bdd44d14"],
+Cell[3726, 121, 9857, 285, 381, "Output",ExpressionUUID->"efae8b97-cac0-4dfb-b6a6-403456514c4f"]
+}, Open ]],
+Cell[13598, 409, 1282, 35, 41, "Input",ExpressionUUID->"c8a865bb-cad1-450c-8187-a809f109c3c8"],
+Cell[CellGroupData[{
+Cell[14905, 448, 1994, 59, 171, InheritFromParent,ExpressionUUID->"bd0a8b22-65f5-4df7-b577-0bf969c52194"],
+Cell[16902, 509, 1870, 56, 70, "Output",ExpressionUUID->"3576d1fd-5496-4abf-b6de-d73b330829ac"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[18809, 570, 757, 16, 24, "Input",ExpressionUUID->"430dc047-855a-4d97-9e15-6a1b8ed47326"],
+Cell[19569, 588, 387, 7, 25, "Output",ExpressionUUID->"4c7ac351-a724-4419-9c0c-ec0428c101c7"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[19993, 600, 1177, 32, 57, "Input",ExpressionUUID->"608d5852-49d9-45d7-9763-cb10da65d830"],
+Cell[21173, 634, 417, 10, 47, "Output",ExpressionUUID->"6e748a34-f946-4d43-8801-0833c28048e5"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[21627, 649, 393, 7, 22, "Input",ExpressionUUID->"5fe4ff16-26ea-47b3-a961-9c776acf99bb"],
+Cell[22023, 658, 319, 5, 25, "Output",ExpressionUUID->"cffc2fe3-b3b9-420d-ae4c-5559b2e114bb"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[22379, 668, 1903, 57, 129, "Input",ExpressionUUID->"c1119876-e977-4ca0-8963-ee017dea27fd"],
+Cell[24285, 727, 752, 21, 55, "Output",ExpressionUUID->"af9d10e0-c1b2-44e5-b2ed-d286376e73b8"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[25074, 753, 227, 4, 22, "Input",ExpressionUUID->"54955d9a-a1b8-49ea-8b22-6f8c2c06fec5"],
+Cell[25304, 759, 171, 2, 25, "Output",ExpressionUUID->"06308438-871f-452c-aef0-07985232fef7"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[25512, 766, 373, 9, 24, "Input",ExpressionUUID->"3b3622a7-fadc-4e42-a347-8de6c9595dd3"],
+Cell[25888, 777, 3799, 115, 282, "Output",ExpressionUUID->"caca42d0-5916-43b6-84ea-cc91bc115d17"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[29724, 897, 570, 11, 24, "Input",ExpressionUUID->"bdc138d2-dd1c-49d9-b891-f23d7c6a21a1"],
+Cell[30297, 910, 8540, 161, 184, "Output",ExpressionUUID->"7dab72fd-cd30-44be-b73a-d23ec16b745a"]
+}, Open ]],
+Cell[CellGroupData[{
+Cell[38874, 1076, 211, 3, 24, "Input",ExpressionUUID->"acd277d9-1867-4c72-bd06-edff0291a14e"],
+Cell[39088, 1081, 156, 2, 25, "Output",ExpressionUUID->"a786f1b5-8e89-47a1-ba0e-8efd965ebaa9"]
+}, Open ]],
+Cell[39259, 1086, 196, 5, 22, "Input",ExpressionUUID->"b3b0287b-1e22-4b15-a4f5-10695aa5f35c"],
+Cell[39458, 1093, 129, 3, 22, "Input",ExpressionUUID->"072b5925-5f74-49be-9a73-34916c763d7c"]
+}
+]
+*)
+
+(* End of internal cache information *)
+