3532 lines
121 KiB
Mathematica
3532 lines
121 KiB
Mathematica
(* 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[ 124133, 3523]
|
|
NotebookOptionsPosition[ 113111, 3255]
|
|
NotebookOutlinePosition[ 113451, 3270]
|
|
CellTagsIndexPosition[ 113408, 3267]
|
|
WindowFrame->Normal*)
|
|
|
|
(* Beginning of Notebook Content *)
|
|
Notebook[{
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "+", "n"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"k", "^", "2"}]}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}]}], "]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]], "Input",
|
|
CellChangeTimes->{{3.714278242480448*^9, 3.714278386121441*^9}, {
|
|
3.7142784362242203`*^9, 3.71427845192585*^9}, {3.714278623956832*^9,
|
|
3.714278766471459*^9}, {3.71427939685915*^9,
|
|
3.714279406485059*^9}},ExpressionUUID->"11f718af-721d-4530-8d00-\
|
|
fc136b151d2a"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"c", "=", "0.1"}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Kappa]", "=", "4"}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"k0", "=", "1"}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"k", "=", "3"}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"k0", " ", "x"}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"-", "2"}], ")"}]}], " ",
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{"I", " ", "k0", " ", "x"}], "]"}], " ", "x", " ",
|
|
RowBox[{"BesselJ", "[",
|
|
RowBox[{"0", ",", " ",
|
|
RowBox[{"k", " ", "x"}]}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-", "c"}], " ", "x"}], "]"}]}], ")"}], "^", "\[Kappa]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}], "Input",
|
|
CellChangeTimes->{{3.714275713710395*^9, 3.714275854008779*^9}, {
|
|
3.714275901119227*^9, 3.7142759040182943`*^9}, 3.7142759848256273`*^9,
|
|
3.7142760852880793`*^9, 3.714279521167551*^9,
|
|
3.714376642974696*^9},ExpressionUUID->"60b62416-1732-4228-8f52-\
|
|
4dd7ea7c260b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.30129960224205`*^-6"}], "+",
|
|
RowBox[{"5.475415847841633`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.71427895383304*^9, 3.7142794125837593`*^9,
|
|
3.71427952256045*^9},ExpressionUUID->"e1c0326f-b1f0-4ebb-9de3-0d92c1b9201a"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"NLimit", "[", " ",
|
|
RowBox[{
|
|
RowBox[{"Hs", "[",
|
|
RowBox[{"q", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}], "]"}],
|
|
",",
|
|
RowBox[{"q", "\[Rule]", "2"}]}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.7142789625057173`*^9, 3.714279028798195*^9}, {
|
|
3.714279076797824*^9, 3.7142790779849653`*^9}, {3.714279432152557*^9,
|
|
3.714279471232699*^9},
|
|
3.714279515514453*^9},ExpressionUUID->"9bf951ee-4aec-4260-9ddf-\
|
|
ddd1330b6243"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"NLimit", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"0.4`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"1.`", " ", "\[ImaginaryI]"}]}], ")"}],
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q"}]], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q"}], "]"}], " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "-", "q"}], "2"], ",",
|
|
FractionBox[
|
|
RowBox[{"3", "-", "q"}], "2"], ",", "1", ",",
|
|
RowBox[{"5.618311533888228`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"5.35077288941736`", " ", "\[ImaginaryI]"}]}]}], "]"}]}], "-",
|
|
RowBox[{"4", " ",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"0.30000000000000004`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"1.`", " ", "\[ImaginaryI]"}]}], ")"}],
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q"}]], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q"}], "]"}], " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "-", "q"}], "2"], ",",
|
|
FractionBox[
|
|
RowBox[{"3", "-", "q"}], "2"], ",", "1", ",",
|
|
RowBox[{"6.893359144853127`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"4.545071963639424`", " ", "\[ImaginaryI]"}]}]}], "]"}]}],
|
|
"+",
|
|
RowBox[{"6", " ",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"0.2`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"1.`", " ", "\[ImaginaryI]"}]}], ")"}],
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q"}]], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q"}], "]"}], " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "-", "q"}], "2"], ",",
|
|
FractionBox[
|
|
RowBox[{"3", "-", "q"}], "2"], ",", "1", ",",
|
|
RowBox[{"7.988165680473372`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"3.328402366863905`", " ", "\[ImaginaryI]"}]}]}], "]"}]}],
|
|
"-",
|
|
RowBox[{"4", " ",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"0.1`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"1.`", " ", "\[ImaginaryI]"}]}], ")"}],
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q"}]], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q"}], "]"}], " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "-", "q"}], "2"], ",",
|
|
FractionBox[
|
|
RowBox[{"3", "-", "q"}], "2"], ",", "1", ",",
|
|
RowBox[{"8.734437800215666`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"1.7645328889324576`", " ", "\[ImaginaryI]"}]}]}], "]"}]}],
|
|
"+",
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"0.`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"1.`", " ", "\[ImaginaryI]"}]}], ")"}],
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q"}]], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q"}], "]"}], " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "-", "q"}], "2"], ",",
|
|
FractionBox[
|
|
RowBox[{"3", "-", "q"}], "2"], ",", "1", ",",
|
|
RowBox[{"9.`", "\[VeryThinSpace]", "+",
|
|
RowBox[{"0.`", " ", "\[ImaginaryI]"}]}]}], "]"}]}]}], ",",
|
|
RowBox[{"q", "\[Rule]", "2"}]}], "]"}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714279029382543*^9, 3.714279078535823*^9, {3.714279413783884*^9,
|
|
3.714279471673715*^9},
|
|
3.7142795169295053`*^9},ExpressionUUID->"d2edfa46-cbb3-4c0a-a2db-\
|
|
5b706afe5c8f"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714279109599843*^9, 3.7142791553425417`*^9}, {
|
|
3.714279476280341*^9, 3.714279528351893*^9}, {3.714280043246078*^9,
|
|
3.7142800433806467`*^9}, 3.714282592755178*^9,
|
|
3.714287530121619*^9},ExpressionUUID->"8a8a5ec5-4164-415c-a832-\
|
|
378df85c3809"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.310140985173348`*^-6"}], "+",
|
|
RowBox[{"5.469690737114341`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{{3.714279116795315*^9, 3.714279156067604*^9},
|
|
3.714279415135985*^9, {3.71427947721791*^9, 3.714279528810555*^9},
|
|
3.714280043970068*^9, 3.71428259343073*^9,
|
|
3.714287530995831*^9},ExpressionUUID->"e6a4d50f-a64a-4c92-bccd-\
|
|
4fd642d8ab43"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs2", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n", "-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}], "\[IndentingNewLine]", "-",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n", "-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"3", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}]}], ")"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]], "Input",
|
|
CellChangeTimes->{
|
|
3.714279590035865*^9, {3.714279690818611*^9, 3.714280013752778*^9}, {
|
|
3.714280457778509*^9,
|
|
3.714280466589781*^9}},ExpressionUUID->"15978efa-9e1d-46b2-b638-\
|
|
930987091033"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs2", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714280030611827*^9, 3.714280038522109*^9},
|
|
3.7142819318470097`*^9,
|
|
3.7142875347922573`*^9},ExpressionUUID->"bd9bb75c-aa2f-4771-820c-\
|
|
573b5d15abe3"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.310138938810269`*^-6"}], "+",
|
|
RowBox[{"5.469690733006516`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.7142874547141237`*^9, 3.714287535713789*^9,
|
|
3.714288543501436*^9},ExpressionUUID->"fdc37902-4c0f-48d5-b1f9-\
|
|
b05ced69d16b"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs20", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}], "\[IndentingNewLine]", "-",
|
|
" ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"3", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}]}], ")"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{
|
|
3.714279590035865*^9, {3.714279690818611*^9, 3.714280013752778*^9}, {
|
|
3.714280457778509*^9, 3.714280466589781*^9}, {3.7142886263091793`*^9,
|
|
3.714288674654853*^9}},ExpressionUUID->"0270ff41-5dea-48e1-8e61-\
|
|
3ad59c8f3c77"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs20", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714280030611827*^9, 3.714280038522109*^9},
|
|
3.7142819318470097`*^9, 3.7142875347922573`*^9,
|
|
3.71428862964427*^9},ExpressionUUID->"dc1bf119-f620-4298-9861-\
|
|
0bebd50af39b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.31013872655064`*^-6"}], "+",
|
|
RowBox[{"5.469690734266986`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.7142874547141237`*^9, 3.714287535713789*^9, 3.714288543501436*^9, {
|
|
3.714288633087429*^9,
|
|
3.714288676731587*^9}},ExpressionUUID->"502ca379-4501-4726-b8a3-\
|
|
a6d4743d21cf"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs3", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{"q", "-", "2"}], ")"}]}], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}],
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], ")"}]}]}], ")"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}]}], "]"}],
|
|
"/",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "+", "n"}], ",",
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], "]"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"Hypergeometric2F1", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"\[Sigma]", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]], "Input",
|
|
CellChangeTimes->{{3.7142805171206827`*^9, 3.7142805173125057`*^9}, {
|
|
3.714281212013117*^9, 3.7142812678804617`*^9}, {3.714281315212207*^9,
|
|
3.71428145933329*^9}},ExpressionUUID->"7bdc1e3b-21a7-4b53-8781-\
|
|
12296dcb7e43"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs3", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714281486641721*^9,
|
|
3.714281487161277*^9}},ExpressionUUID->"e99ec6cf-d99f-42bb-b94e-\
|
|
4c6feeaefe2f"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"0.0005961271568821758`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"0.0005802053412378161`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.7142814876849213`*^9},ExpressionUUID->"9defb7fc-3a2a-4372-b667-\
|
|
e589fcb6d801"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs4", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"k", "^", "n"}], " ", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n"}], "]"}]}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"Hypergeometric2F1", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "+", "n"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"k", "^", "2"}]}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}]}], "]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]], "Input",
|
|
CellChangeTimes->{{3.714281823698204*^9, 3.71428186208356*^9},
|
|
3.714281944086228*^9},ExpressionUUID->"c903db13-544e-45bc-b87f-\
|
|
4a3b87557ce7"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs4", "[",
|
|
RowBox[{"2.001", ",", "2", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{3.7142818741710377`*^9,
|
|
3.714281937081643*^9},ExpressionUUID->"c75220ed-031f-49e5-9467-\
|
|
d46ac941cb11"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"3.3634365526058474`*^-6", "-",
|
|
RowBox[{"8.613661063167077`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714281874941059*^9, {3.71428193770849*^9,
|
|
3.71428194777015*^9}},ExpressionUUID->"2ca46fac-dce2-4111-be04-\
|
|
222479370c7c"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs5", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{"q", "-", "2"}], ")"}]}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/", "\[IndentingNewLine]",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}],
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], ")"}]}], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n"}], "]"}]}], ")"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}]}], "]"}],
|
|
"/",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "+", "n"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}]}], "]"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"Hypergeometric2F1", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]], "Input",
|
|
CellChangeTimes->{
|
|
3.714282098516695*^9, {3.714282130447603*^9, 3.7142821401345367`*^9}, {
|
|
3.714282216920789*^9, 3.7142823384310217`*^9}, {3.7142824377107773`*^9,
|
|
3.714282439156217*^9}, {3.714282477583288*^9,
|
|
3.7142824926773233`*^9}},ExpressionUUID->"91fb5021-708e-4953-92bf-\
|
|
e65236c1db92"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs5", "[",
|
|
RowBox[{"2.000", ",", "2", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{3.714282348723709*^9,
|
|
3.714282689741394*^9},ExpressionUUID->"1d288674-7aa8-40c7-ad24-\
|
|
caf62a39a091"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "0.013915081934793683`"}], "-",
|
|
RowBox[{"0.003057622531865256`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.7142823493744793`*^9, 3.714282440942499*^9,
|
|
3.714282494600651*^9,
|
|
3.714282690480345*^9},ExpressionUUID->"d5266c94-5c79-4a80-a294-\
|
|
09b967a31d30"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs6", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}], " ",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}],
|
|
RowBox[{"\[Pi]", "/", "\[IndentingNewLine]",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}]}], " ", ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", " ", "/", " ",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}],
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}],
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}], " ",
|
|
RowBox[{"s", "!"}]}], ")"}]}]}], "\[IndentingNewLine]", "-",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", " ", "/", " ",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}],
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}],
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}], " ",
|
|
RowBox[{"s", "!"}]}], ")"}]}], "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"-", "\[Sigma]"}], " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}]}]}],
|
|
"\[IndentingNewLine]", "\[IndentingNewLine]", ")"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}], "]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}]], "Input",
|
|
CellChangeTimes->{{3.714286486428625*^9, 3.714286510862175*^9}, {
|
|
3.714286599838471*^9, 3.7142866149002047`*^9}, {3.7142866461705503`*^9,
|
|
3.714286660781233*^9}, {3.714286693444262*^9, 3.7142866951275806`*^9}, {
|
|
3.714286729608942*^9, 3.714286737178653*^9}, {3.714286770356913*^9,
|
|
3.714286967963607*^9}},ExpressionUUID->"35e0146b-a88e-4958-9a81-\
|
|
4af639dd5abb"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs6", "[",
|
|
RowBox[{"2.1", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714286986468603*^9, 3.714286989151475*^9}, {
|
|
3.714287136219614*^9,
|
|
3.714287149633726*^9}},ExpressionUUID->"ee4e143e-b867-4930-86b6-\
|
|
de913836ba38"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"8.847276454415696`*^-15", "-",
|
|
RowBox[{"5.680986655414175`*^-16", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714286989900689*^9, {3.714287138803092*^9,
|
|
3.7142871522134933`*^9}},ExpressionUUID->"e7215961-ce53-448f-a9e3-\
|
|
0eda8cc30a4d"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs21", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"1", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}], "\[IndentingNewLine]", "-",
|
|
" ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Hypergeometric2F1Regularized", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
RowBox[{"3", "/", "2"}], ",",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}]}], "]"}]}]}], ")"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{
|
|
3.714279590035865*^9, {3.714279690818611*^9, 3.714280013752778*^9}, {
|
|
3.714280457778509*^9, 3.714280466589781*^9}, 3.714287233376712*^9, {
|
|
3.7142872781308537`*^9, 3.7142873155616083`*^9}, {3.714287408146644*^9,
|
|
3.714287430524211*^9}, {3.7142883819535217`*^9, 3.714288384252969*^9}, {
|
|
3.714288757815189*^9, 3.714288776382951*^9}, {3.714288865962365*^9,
|
|
3.7142888667854*^9}},ExpressionUUID->"a95bd250-d811-48c2-87a5-\
|
|
6e237d1225d7"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs21", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714280030611827*^9, 3.714280038522109*^9},
|
|
3.7142819318470097`*^9, 3.714287234837658*^9, 3.7142874635342197`*^9,
|
|
3.714288808579112*^9},ExpressionUUID->"2ea68715-88f5-4465-af14-\
|
|
0b3fcd58d31b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.31013872655064`*^-6"}], "+",
|
|
RowBox[{"5.469690734266986`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{{3.714280031311132*^9, 3.714280039292564*^9},
|
|
3.714280469107683*^9, 3.714281932658917*^9, 3.71428257860351*^9, {
|
|
3.714287436217915*^9, 3.714287464439518*^9}, 3.714287540141326*^9,
|
|
3.714288385713441*^9, 3.714288548425488*^9, {3.714288760252366*^9,
|
|
3.71428877881402*^9}, 3.714288809246872*^9,
|
|
3.714288869829924*^9},ExpressionUUID->"a7f32ea2-344b-47e7-80d7-\
|
|
9ffb74f31a08"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs3a", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"Sum", "[", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "\[IndentingNewLine]", "\t",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}], ")"}], "^", "s"}]}],
|
|
"\[IndentingNewLine]", "-", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "\[IndentingNewLine]", "\t",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}], ")"}], "^", "s"}]}]}], ",",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.7142893672643137`*^9, 3.714289382815003*^9}, {
|
|
3.7142894295982018`*^9, 3.714289459307962*^9}, {3.714289498065723*^9,
|
|
3.714289626198709*^9}, {3.714289980275403*^9, 3.714290000381897*^9}, {
|
|
3.7142903246351557`*^9, 3.714290331819459*^9}, {3.714290378125929*^9,
|
|
3.714290401543976*^9}},ExpressionUUID->"3d8bffca-d6ac-482a-b1a2-\
|
|
b2e45fd6aca8"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs3a", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.7142896421583853`*^9, 3.714289642998307*^9}, {
|
|
3.714289675624762*^9, 3.714289675800159*^9},
|
|
3.714289898236577*^9},ExpressionUUID->"1abff5df-c88c-4949-93d7-\
|
|
b00aaf2d034c"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.310138190814039`*^-6"}], "+",
|
|
RowBox[{"5.4696907326974446`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.714289734734824*^9, 3.714289825697638*^9,
|
|
3.714289971009247*^9, 3.7142902647769127`*^9,
|
|
3.714290462593569*^9},ExpressionUUID->"69027225-5785-4f7e-9ff2-\
|
|
0a3cee3e176d"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"//",
|
|
RowBox[{"oto\[CHacek]en\[EAcute]", " ", "po\[RHacek]ad\[IAcute]", " ",
|
|
RowBox[{"sumy", ":"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714290586711878*^9,
|
|
3.714290599100697*^9}},ExpressionUUID->"d5926fc0-c5a2-4b56-894e-\
|
|
116ee99824aa"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs3b", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[", " ",
|
|
RowBox[{
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}],
|
|
"^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "\[IndentingNewLine]", "\t",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}], ")"}], "^", "s"}]}],
|
|
"\[IndentingNewLine]", "-", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k", "^", "2"}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}],
|
|
"^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}]}], "/", "2"}],
|
|
")"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "\[IndentingNewLine]", "\t",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], "/",
|
|
RowBox[{"k", "^", "2"}]}], ")"}], "^", "s"}]}]}],
|
|
"\[IndentingNewLine]", ")"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}], "]"}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.7142893672643137`*^9, 3.714289382815003*^9}, {
|
|
3.7142894295982018`*^9, 3.714289459307962*^9}, {3.714289498065723*^9,
|
|
3.714289626198709*^9}, {3.714289980275403*^9, 3.714290000381897*^9}, {
|
|
3.714290037095689*^9, 3.714290082714675*^9}, {3.714290420096354*^9,
|
|
3.7142904284034443`*^9}, {3.714290545669217*^9,
|
|
3.7142905464623137`*^9}},ExpressionUUID->"5e62d660-2944-47ca-8ee0-\
|
|
e6a553ed7d63"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs3b", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.7142896421583853`*^9, 3.714289642998307*^9}, {
|
|
3.714289675624762*^9, 3.714289675800159*^9}, 3.714289898236577*^9,
|
|
3.7142900408710833`*^9},ExpressionUUID->"ac03b840-c27b-4fe3-bb0b-\
|
|
82daba30770b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3101401399657503`*^-6"}], "+",
|
|
RowBox[{"5.469690735905011`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.7142903189715757`*^9, 3.714290502891795*^9,
|
|
3.7142906073604593`*^9},ExpressionUUID->"e5d1f184-d264-4d0a-af52-\
|
|
34362b04d2b6"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs4a", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"k", "^", "n"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], " ", "-", " ",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q", "-", "n", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n", "+",
|
|
RowBox[{"2", "s"}]}], ")"}]}]}], "\[IndentingNewLine]", "-",
|
|
" ",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "3"}], "+", "q", "-", "n", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n", "+",
|
|
RowBox[{"2", "s"}]}], ")"}]}]}]}], "\[IndentingNewLine]",
|
|
")"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.7142893672643137`*^9, 3.714289382815003*^9}, {
|
|
3.7142894295982018`*^9, 3.714289459307962*^9}, {3.714289498065723*^9,
|
|
3.714289626198709*^9}, {3.714289980275403*^9, 3.714290000381897*^9}, {
|
|
3.7142903246351557`*^9, 3.714290331819459*^9}, {3.714290378125929*^9,
|
|
3.714290401543976*^9}, {3.71429065707089*^9, 3.714290657857644*^9}, {
|
|
3.714290726660638*^9, 3.714290781397086*^9}, {3.714290812028606*^9,
|
|
3.7142909334348288`*^9}},ExpressionUUID->"4ab261b6-0c49-491c-9869-\
|
|
75df5adc999d"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs4a", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714290670918292*^9, 3.7142906716436872`*^9},
|
|
3.714291273201295*^9},ExpressionUUID->"28a7cebd-2387-4639-b462-\
|
|
54326a81706f"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3101399766027096`*^-6"}], "+",
|
|
RowBox[{"5.4696907340925925`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714290941279868*^9},ExpressionUUID->"408303d5-6eab-465e-b43c-\
|
|
43bc1a3b6810"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs4b", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/", "\[IndentingNewLine]",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}]}], ")"}]}], "\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "s"}], ")"}]}]}], "\[IndentingNewLine]", "-", " ",
|
|
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]", " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "3"}], "+", "q", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"2", "s"}]}], ")"}]}]}]}], "\[IndentingNewLine]",
|
|
")"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.7142893672643137`*^9, 3.714289382815003*^9}, {
|
|
3.7142894295982018`*^9, 3.714289459307962*^9}, {3.714289498065723*^9,
|
|
3.714289626198709*^9}, {3.714289980275403*^9, 3.714290000381897*^9}, {
|
|
3.7142903246351557`*^9, 3.714290331819459*^9}, {3.714290378125929*^9,
|
|
3.714290401543976*^9}, {3.71429065707089*^9, 3.714290657857644*^9}, {
|
|
3.714290726660638*^9, 3.714290781397086*^9}, {3.714290812028606*^9,
|
|
3.7142909334348288`*^9}, {3.714291024637805*^9, 3.7142912095547323`*^9},
|
|
3.714291266458826*^9},ExpressionUUID->"a0aeba62-9dcc-4486-a26f-\
|
|
5ef44b8b7a21"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs4b", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714291278032549*^9,
|
|
3.7142912786096277`*^9}},ExpressionUUID->"959e5694-67a4-4fb4-a01a-\
|
|
51c9949a29f0"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.310140512339311`*^-6"}], "+",
|
|
RowBox[{"5.469690733307822`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714291278895607*^9},ExpressionUUID->"b6a40e5c-36ec-408f-a622-\
|
|
54079303243a"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs4c", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/", "\[IndentingNewLine]",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}]}], ")"}]}], "\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "s"}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}]}], ")"}],
|
|
"\[IndentingNewLine]", "-", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}],
|
|
"]"}]}]}], ")"}], "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}]}]}],
|
|
"\[IndentingNewLine]", ")"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.7142893672643137`*^9, 3.714289382815003*^9}, {
|
|
3.7142894295982018`*^9, 3.714289459307962*^9}, {3.714289498065723*^9,
|
|
3.714289626198709*^9}, {3.714289980275403*^9, 3.714290000381897*^9}, {
|
|
3.7142903246351557`*^9, 3.714290331819459*^9}, {3.714290378125929*^9,
|
|
3.714290401543976*^9}, {3.71429065707089*^9, 3.714290657857644*^9}, {
|
|
3.714290726660638*^9, 3.714290781397086*^9}, {3.714290812028606*^9,
|
|
3.7142909334348288`*^9}, {3.714291024637805*^9, 3.7142912095547323`*^9},
|
|
3.714291266458826*^9, {3.714291313707507*^9, 3.714291314301175*^9}, {
|
|
3.714291530771325*^9, 3.7142916010936403`*^9}, {3.71429179054211*^9,
|
|
3.714291821418215*^9}},ExpressionUUID->"811cd081-362a-41a3-bd5d-\
|
|
f77c96ca788c"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs4c", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.71429161488424*^9,
|
|
3.714291615400077*^9}},ExpressionUUID->"dc6edbc3-0394-4cd5-be4f-\
|
|
23c469fcc0bb"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3101419409702477`*^-6"}], "+",
|
|
RowBox[{"5.4696907333950185`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.714291617372395*^9,
|
|
3.714291829663458*^9},ExpressionUUID->"d1aa622a-f7c6-4b70-a80c-\
|
|
c09426523c9d"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"Hs4d", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/", "\[IndentingNewLine]",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}]}], ")"}]}], "\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "s"}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n"}], ")"}], "/", "2"}], "]"}]}]}], ")"}],
|
|
"\[IndentingNewLine]", "-", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}],
|
|
"]"}]}]}], ")"}], "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}]}]}],
|
|
"\[IndentingNewLine]", ")"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",",
|
|
RowBox[{"Ceiling", "[",
|
|
RowBox[{"\[Kappa]", "/", "2"}], "]"}], ",", "\[Infinity]"}],
|
|
"}"}]}], "]"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714291934746553*^9, 3.7142919359793797`*^9}, {
|
|
3.714291966225366*^9, 3.714291969149551*^9}, {3.714296445383353*^9,
|
|
3.7142964457053547`*^9}, 3.714296507945154*^9, {3.71429655564856*^9,
|
|
3.714296567006785*^9},
|
|
3.714304904564191*^9},ExpressionUUID->"747e5737-fbc4-443f-8253-\
|
|
54d4a139ee63"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Hs4d", "[",
|
|
RowBox[{"2.00001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{3.7142964324128847`*^9, 3.7143050627984667`*^9,
|
|
3.7143052251049547`*^9},ExpressionUUID->"69f1a251-919e-4400-b0c2-\
|
|
e2d5db4dccff"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3014227810251192`*^-6"}], "+",
|
|
RowBox[{"5.4753586632338605`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{{3.714296432738432*^9, 3.714296451646615*^9},
|
|
3.714296512203021*^9, 3.7142965693207006`*^9, 3.714305064845126*^9,
|
|
3.714305227033358*^9},ExpressionUUID->"8b2e58e7-5d5a-4697-824a-\
|
|
0a411cb068bf"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"s", "+",
|
|
RowBox[{"1", "/", "2"}]}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{"s", "+",
|
|
RowBox[{"1", "/", "2"}]}], ")"}]}], "/",
|
|
RowBox[{"s", "!"}]}],
|
|
RowBox[{"x", "^", "s"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.714302231995471*^9, 3.714302249139989*^9}, {
|
|
3.714302307582074*^9, 3.714302355910602*^9}, 3.714302660798676*^9, {
|
|
3.714302710981365*^9,
|
|
3.714302719324584*^9}},ExpressionUUID->"c015003f-0699-49e2-b866-\
|
|
714eb92d5c6f"],
|
|
|
|
Cell[BoxData[
|
|
FractionBox[
|
|
RowBox[{"2", " ",
|
|
SqrtBox["\[Pi]"], " ",
|
|
RowBox[{"ArcSinh", "[",
|
|
SqrtBox["x"], "]"}]}],
|
|
SqrtBox["x"]]], "Output",
|
|
CellChangeTimes->{
|
|
3.714302357238738*^9, {3.7143027140973*^9,
|
|
3.7143027212700253`*^9}},ExpressionUUID->"b5efa070-aee7-4251-898d-\
|
|
4ad75f8ee655"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "/", "2"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.7143026654616632`*^9,
|
|
3.7143026672066393`*^9}},ExpressionUUID->"65bc3149-1111-4cfb-a506-\
|
|
bc566f00020a"],
|
|
|
|
Cell[BoxData[
|
|
SqrtBox["\[Pi]"]], "Output",
|
|
CellChangeTimes->{
|
|
3.7143026674794493`*^9},ExpressionUUID->"385fc41d-508e-4bcc-87ce-\
|
|
366fda2afa16"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Plot", "[",
|
|
RowBox[{
|
|
RowBox[{"ArcSinh", "[",
|
|
RowBox[{"1", "/", "x"}], "]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "0", ",", "200"}], "}"}]}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.714303154749318*^9, 3.7143031551135693`*^9}, {
|
|
3.714303200349794*^9,
|
|
3.7143032278424473`*^9}},ExpressionUUID->"fbdde755-24a3-46b6-b157-\
|
|
f28e6b6a8022"],
|
|
|
|
Cell[BoxData[
|
|
GraphicsBox[{{{}, {},
|
|
TagBox[
|
|
{RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
|
|
1.], LineBox[CompressedData["
|
|
1:eJwVz3c81Ysfx3FJqdRNUri53SRyrOwkfD6OERlZld0x4iuEfO2jDoVEkXHQ
|
|
yco69qqEFG5SKisqSaUk67pGGZX8/P54P17/PP95izh5m5/m5ODgiFzd/9tr
|
|
BVpyJnIYckbmQLxjBXB9kPPsNJVDN58JZbpdBdy65tbulSyHz8a65++/K4cP
|
|
//ZGFe+SR/ukJwV/jJWBdWnFiqikAg5eVdwpLl0KaKWRynNYAWc+Saps4SsF
|
|
Ca5nsnOGCji+aZC+b7EEFm1H7Fu8FNB7zN9r6FEJMHn+uk+rUsBLiz+eR9FK
|
|
4OWZK0EZqopYHeLez59dDMaU0zM7jyjhgrvA0mGNIvgR698weEIJ9Wc2YSml
|
|
CNhTkZF5rkrYRx94oLazCLhuFwgpRCmhCD4xrvqPDfWao1rGrUq4/fcIdTmX
|
|
DeKWHokXtZXxqoBoxettbOC44Ks4DSpo2yft+52jEMqHwpfvHlPBlkIT4dvT
|
|
BWCrk9gWdkoF2+McymM+FsDdjTW2PBdUsHVbx+OgpgLwTP52UfyBCvJmNtxy
|
|
DC+A/qLAXjv1gzj4TCZIhbsAqnvD/J+qqiI75zb1iUg+sCNdnD4ZqOLQrwzL
|
|
cb58yFQxPPbTRhXfN4driXDlw5U0QYpMmCoOxlUuPx/JA2e7mncJzao4sEHf
|
|
UqAsD/iHR7Wtjh5CynJSuoVGHgTMWfB9tVXDpPOh+o4eueCZp7ay4qmG04kJ
|
|
tj0OueB0XGRS8LwanhU1yLEwzwWT2qnWo9lqaPlu9kjUoVzYHxoTXD6shk5v
|
|
M4rZG3LhDefDj/5eh7E2t5LzSdEtOMwnWcF1QR1dbSSdquZzoGONzdMX8eo4
|
|
SzMcFJ3MAdpMzGdmtjoWG8TdvTWUA5e6xgQoLeo48/GDdOvzVX+tiGG8TgPr
|
|
6wf7buSueh6KOTNWA49dW/o70GLVc0l835+miZv7rLi9GrJh57eTW2fYmph+
|
|
0LRdpzob2J+jKfV1mii2QfeqeFE2vGj5am80oIka0RPd3KmrPrzwsc9uwOxm
|
|
9RTwy4aiZfG0ujxAa44VjwCZbHD7mZiRb4W4/7m1vopZFiSbTP4TTEPssHZc
|
|
0KRmQVOO7rgxgbix8PPkccUsENRfVJkPROSMmLcu35EFT5Ltu/RSEdNEN/Nw
|
|
D2QCRVZi7WgfYp7jf/f0iEyYoDUSkhZa+OB7wK+wuAzwfvxVscKYipXyhyIp
|
|
iyxowlIP6RNUrJWeDzw4wQLeBp/cIgcqFjGSaGbvWVBZscSX501Fh0VaN/sR
|
|
C2bTeGbTrlMxPT/Mffg6CwI8DlRG9FFx/nVFCkOWBXTeQOkT9tqYR7mT9dn7
|
|
BlyxWS+27KmDK22HGHa86aApt3l+jNTBnoVhLFyfDjPr+Npe0XWwWKri4tKv
|
|
NLCq+su9Mk4HKzqHPW+PpYHYBuUy51IddM5hDUb8kwZNd5yV2yd0kPX4j6GQ
|
|
oDSY523WTT2ji29CnahZX1PBuS3UVZ7Qw32ZZZp/9TPBRV3cZN5bD5VCR7WV
|
|
upngWtWl3BCoh75n1ZtNnzKBYImt043Ww7az8Vtu1DHhrHdnrlWBHl6WUDT1
|
|
usGEEAHRoQvDeviv/UjlNnsmJLq123bQjuCmLs0f0iMp0MwtZOZhrY+R/bXx
|
|
N9engPw2BnOMPIrq8v0W49ZJQJvTkDbxNMLRRw/M1NYmQk5mq/DCVRO0De4R
|
|
Tm5OgJpragM95qbY8kjf3dw+Hsh6DdMMIzPUHaLpBaVdhYft+bMxtub4wYjB
|
|
uPc5FtyPHGp0t7JArlB6hGzgFbjJlr/jccYSHa1O5f/eHgPX5xrDfQKPoy6v
|
|
1taG/mgYKfYPIM6dQN3guOv8nVHwSC8xLSLiJD7jZbYItUXCmmNvP8UzrPBq
|
|
pqS0yZdLwKF1d012sDVO3XdSXid0CS4LLBj3RNtg9ZSfoJXRRfCxcW5ZuGiL
|
|
1MtU/nBWBAyFK7lAih36maW03hgLB/6k9LcxOfZYmqE14nEkHAp4JYPbWA6Y
|
|
3rs0LlHPgOZzf+vuLzmFPKn2615KMiBzT7awhiUNX/kZvX1lcwEa6T84iC4a
|
|
3vMYHzb0Ow8ihty/RE86okuPcnNNVBjc7qTE9vc6Yqb6z764u3SIe/xbbo+9
|
|
E+pKtg8HD4VCkt/YGdo7JzT4Mqd/SiwU6Dt2b91r7Yz8vyaYJwJCoIiyrp38
|
|
6Iys5svdsi3BsP7bGqFmBxfUFdtnE7I3GArlqs5nfnXBZNmy8YmwIKA4FZss
|
|
e53GtvREXpvhQBgwqKs+PnEaRdayd9fbBkJPbogM+LlipzdXullHACzWmKnv
|
|
nXXFT3tWxpRMA2Dl3rk7UWfdUGFNyL2mfn8wlNPTpX93w671A/UlDv7QZn/A
|
|
3cOPwLqNjQasaRL4nImGDH8CA3c1faqbIsGByNnSFUhg45x8z5tJEubPba9W
|
|
ohPoMMl1W2CMhH2XF34sXyJwB+M9I32IBEb1w7gEJoF95AGpsh4S1LhNK+/U
|
|
ETgp4buX/w4JUZtjOEcbCNwj9dpOo4aE7m0tln8+IHBB7eW4axUJhLDS0vmW
|
|
VR+ttKahjIRUeUGq/jMCd1fI7nArIOGb7ceXb98RKFhW9bgjlQR0FBLf8oHA
|
|
MONdcsspJMS6mgfBEIHB6fUTUskk7PV9JJz3hUBtKW3rKwkkmEWxT3tNETiT
|
|
J8xjGEPCzdih2qxpAkffvlxLjyZhNOHPTT2zBMb6bjcpi1z9w4orV1kgUKS0
|
|
7xVvBAnPs1s53JcIzLWS2aHNIEGg4Lc56yeB0wG1ef7nSXAqOZj/YplAqUyH
|
|
SDadhPJKn4WVFQI5vbbUDoSQ8D/LoOWh
|
|
"]]},
|
|
Annotation[#, "Charting`Private`Tag$637069#1"]& ]}, {}, {}},
|
|
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
|
|
Axes->{True, True},
|
|
AxesLabel->{None, None},
|
|
AxesOrigin->{0, 0.0049999792689405815`},
|
|
DisplayFunction->Identity,
|
|
Frame->{{False, False}, {False, False}},
|
|
FrameLabel->{{None, None}, {None, None}},
|
|
FrameTicks->{{Automatic,
|
|
Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic,
|
|
Charting`ScaledFrameTicks[{Identity, Identity}]}},
|
|
GridLines->{None, None},
|
|
GridLinesStyle->Directive[
|
|
GrayLevel[0.5, 0.4]],
|
|
ImagePadding->All,
|
|
Method->{
|
|
"DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" ->
|
|
AbsolutePointSize[6], "ScalingFunctions" -> None,
|
|
"CoordinatesToolOptions" -> {"DisplayFunction" -> ({
|
|
(Identity[#]& )[
|
|
Part[#, 1]],
|
|
(Identity[#]& )[
|
|
Part[#, 2]]}& ), "CopiedValueFunction" -> ({
|
|
(Identity[#]& )[
|
|
Part[#, 1]],
|
|
(Identity[#]& )[
|
|
Part[#, 2]]}& )}},
|
|
PlotRange->{{0, 200}, {0.0049999792689405815`, 0.06129119349380559}},
|
|
PlotRangeClipping->True,
|
|
PlotRangePadding->{{
|
|
Scaled[0.02],
|
|
Scaled[0.02]}, {
|
|
Scaled[0.05],
|
|
Scaled[0.05]}},
|
|
Ticks->{Automatic, Automatic}]], "Output",
|
|
CellChangeTimes->{{3.714303214207242*^9,
|
|
3.7143032295013943`*^9}},ExpressionUUID->"f3bbc0c5-f6fc-4fd8-ade3-\
|
|
1015c944f07e"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSq2n0a", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/", "\[IndentingNewLine]",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "^", "n"}], " ",
|
|
RowBox[{"k0", "^", "q"}]}], ")"}]}], "\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "+", "q", "-",
|
|
RowBox[{"2", "s"}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "s"}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"-", " ",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "-", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}],
|
|
"]"}]}]}], ")"}]}], "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}]}],
|
|
"\[IndentingNewLine]", ")"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",",
|
|
RowBox[{"Ceiling", "[",
|
|
RowBox[{"\[Kappa]", "/", "2"}], "]"}], ",", "\[Infinity]"}],
|
|
"}"}]}], "]"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714304912719702*^9, 3.714304968618021*^9},
|
|
3.7143050430084667`*^9,
|
|
3.7143051086652327`*^9},ExpressionUUID->"e139a925-f883-4746-b86a-\
|
|
c2602253404c"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0a", "[",
|
|
RowBox[{"2.00001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714304973552258*^9, 3.7143050024644413`*^9},
|
|
3.714305067340577*^9, 3.7143051099639606`*^9,
|
|
3.714305218174539*^9},ExpressionUUID->"7c490700-b77f-4e41-bc26-\
|
|
5314e7de681a"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.301284530022639`*^-6"}], "+",
|
|
RowBox[{"5.475391561810337`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714304974643896*^9, {3.714305044802492*^9, 3.714305068681655*^9},
|
|
3.714305112750959*^9,
|
|
3.714305219521511*^9},ExpressionUUID->"2ec8c9bc-264b-4c99-966f-\
|
|
0e16399c5a47"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSq2n0b", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=", " ",
|
|
RowBox[{"\[Pi]", " ",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"2", "-", "q", "+", "n"}], "]"}], "/", "\[IndentingNewLine]",
|
|
|
|
RowBox[{"(",
|
|
RowBox[{"k0", "^", "2"}], ")"}]}], "\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "s"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "s"}], ")"}]}],
|
|
RowBox[{"(", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"-", " ",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], ",", "s"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], ",", "s"}], "]"}],
|
|
"\[IndentingNewLine]", "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"3", "/", "2"}], "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}],
|
|
"]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"q", "+", "n", "-", "1"}], ")"}], "/", "2"}],
|
|
"]"}]}]}], ")"}]}], "\[IndentingNewLine]", " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}]}],
|
|
"\[IndentingNewLine]", ")"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",",
|
|
RowBox[{"Ceiling", "[",
|
|
RowBox[{"\[Kappa]", "/", "2"}], "]"}], ",", "\[Infinity]"}],
|
|
"}"}]}], "]"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714304912719702*^9, 3.714304968618021*^9},
|
|
3.7143050430084667`*^9, {3.714305120462102*^9, 3.714305193734703*^9}, {
|
|
3.714305245188303*^9, 3.714305261703126*^9}, {3.714305292783976*^9,
|
|
3.714305309384357*^9}, 3.7143053988421583`*^9, 3.714305635760743*^9, {
|
|
3.714306069224308*^9,
|
|
3.714306090556497*^9}},ExpressionUUID->"03b0ced6-4edd-459d-b511-\
|
|
4346ebf0b6b3"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0b", "[",
|
|
RowBox[{"2.00001", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714304973552258*^9, 3.7143050024644413`*^9},
|
|
3.714305067340577*^9, 3.714305123688612*^9,
|
|
3.714305212337887*^9},ExpressionUUID->"56fbb4e1-8f74-4c7e-a055-\
|
|
064f55e0a5e4"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3013287572355085`*^-6"}], "+",
|
|
RowBox[{"5.475485310843043`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714304974643896*^9, {3.714305044802492*^9, 3.714305068681655*^9},
|
|
3.714305144735292*^9, {3.714305175923616*^9, 3.714305214276556*^9}, {
|
|
3.71430608320949*^9,
|
|
3.714306092338118*^9}},ExpressionUUID->"ccc74805-25fc-4b7a-92ea-\
|
|
ecc47f1289c3"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSpecq2n0f", "[",
|
|
RowBox[{"\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"Sqrt", "[", "\[Pi]", "]"}]}],
|
|
RowBox[{"k0", "^",
|
|
RowBox[{"(",
|
|
RowBox[{"-", "2"}], ")"}]}],
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{"-", "2"}], ")"}]}],
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}], "\[IndentingNewLine]",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}], " ",
|
|
RowBox[{"ArcSinh", "[",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}], ")"}], "]"}]}], ",",
|
|
" ",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.7143046021467648`*^9, 3.7143047956060467`*^9}, {
|
|
3.714304881641884*^9,
|
|
3.714304885767686*^9}},ExpressionUUID->"fdc5165f-652c-4553-b7d6-\
|
|
e673dd5aaf0c"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSpecq2nf", "[",
|
|
RowBox[{"4", ",", "0.1", ",", "1", ",", "3"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.7143048200786963`*^9, 3.714304826231525*^9},
|
|
3.714304888515621*^9},ExpressionUUID->"14bbf919-b4de-4582-b295-\
|
|
f820ec4e1863"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"7.331304077260176`*^-6", "-",
|
|
RowBox[{"0.000019758518864176108`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.7143048268370647`*^9},ExpressionUUID->"6f771c74-3b2b-4814-8148-\
|
|
354f0c429799"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSq2n0c", "[",
|
|
RowBox[{"\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}], "]"}], ":=",
|
|
RowBox[{"-",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"1", "/",
|
|
RowBox[{"(",
|
|
RowBox[{"2",
|
|
RowBox[{"k0", "^", "2"}]}], ")"}]}],
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"k", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"-", "2"}], "s"}], "-", "1"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "s"}], "+", "1"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Sqrt", "[", "\[Pi]", "]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], ")"}],
|
|
RowBox[{"s", "!"}]}], ")"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714305404306922*^9, 3.714305411057556*^9}, {
|
|
3.714305459942329*^9, 3.714305642256468*^9}, {3.714305723102551*^9,
|
|
3.714305821193959*^9}, {3.714305899344687*^9, 3.714305911930727*^9}, {
|
|
3.7143061445366173`*^9, 3.714306145331019*^9}, {3.714306226374571*^9,
|
|
3.714306240825282*^9}, {3.714306276550058*^9,
|
|
3.714306281792178*^9}},ExpressionUUID->"4fe8906e-3924-4127-922e-\
|
|
cba66e63b281"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0c", "[",
|
|
RowBox[{"4", ",", "0.1", ",", "1", ",", "3"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.714305847713162*^9, 3.714305852616592*^9}, {
|
|
3.7143061491190643`*^9,
|
|
3.714306149598337*^9}},ExpressionUUID->"314b1bc8-d2f0-41e7-a76a-\
|
|
821f0a313b1d"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3012995221394217`*^-6"}], "+",
|
|
RowBox[{"5.475415753075641`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714305853642666*^9, {3.714305901201582*^9, 3.7143059145741863`*^9},
|
|
3.714306025296417*^9, 3.714306151971169*^9, 3.714306242862033*^9,
|
|
3.7143062840848217`*^9},ExpressionUUID->"a7a1768d-8381-468e-9182-\
|
|
cad6264fbeb4"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSq2n0d", "[",
|
|
RowBox[{"\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}], "]"}], ":=",
|
|
RowBox[{"-",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"1", "/",
|
|
RowBox[{"(",
|
|
RowBox[{"2",
|
|
RowBox[{"k0", "^", "2"}]}], ")"}]}],
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "s"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Sqrt", "[", "\[Pi]", "]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], ")"}],
|
|
RowBox[{"s", "!"}]}], ")"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714306470931181*^9,
|
|
3.714306530662891*^9}},ExpressionUUID->"1b478334-5bf6-4ef1-a489-\
|
|
1d672116aa7a"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0d", "[",
|
|
RowBox[{"4", ",", "0.1", ",", "1", ",", "3"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.71430647467421*^9,
|
|
3.71430647483814*^9}},ExpressionUUID->"e3c36c04-d161-4900-98a4-\
|
|
33e7dc51e872"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3012995221394217`*^-6"}], "+",
|
|
RowBox[{"5.475415753075641`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714306532722279*^9},ExpressionUUID->"fed7bff7-9bbc-40e2-8c1c-\
|
|
b16c41ea5128"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSq2n0e", "[",
|
|
RowBox[{"\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}], "]"}], ":=",
|
|
RowBox[{"-",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"1", "/",
|
|
RowBox[{"(",
|
|
RowBox[{"2",
|
|
RowBox[{"k0", "^", "2"}]}], ")"}]}],
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "s"}], "+", "1"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Sqrt", "[", "\[Pi]", "]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], ")"}],
|
|
RowBox[{"s", "!"}]}], ")"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714306470931181*^9, 3.714306530662891*^9},
|
|
3.714306564507558*^9, {3.714306624898449*^9, 3.7143066749187202`*^9}, {
|
|
3.714306734732802*^9,
|
|
3.714306738819399*^9}},ExpressionUUID->"5d9b2d82-846d-4549-b840-\
|
|
40cefa18d8c0"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0e", "[",
|
|
RowBox[{"4", ",", "0.1", ",", "1", ",", "3"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.71430647467421*^9, 3.71430647483814*^9},
|
|
3.714306561660407*^9, {3.714306701599392*^9,
|
|
3.7143067030037622`*^9}},ExpressionUUID->"d9d5283f-d204-4f19-83d0-\
|
|
7773fcf3cd30"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3012995221394217`*^-6"}], "+",
|
|
RowBox[{"5.475415753075641`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714306532722279*^9, {3.714306715505674*^9,
|
|
3.714306740900689*^9}},ExpressionUUID->"09894aad-397c-465b-b0aa-\
|
|
c9f601a9aec6"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsSq2n0f", "[",
|
|
RowBox[{"\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}], "]"}], ":=",
|
|
RowBox[{"-",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{"1", "/",
|
|
RowBox[{"k0", "^", "2"}]}],
|
|
RowBox[{"ArcSinh", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "/", "k"}], "]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714306470931181*^9, 3.714306530662891*^9},
|
|
3.714306564507558*^9, {3.714306624898449*^9, 3.7143066749187202`*^9}, {
|
|
3.714306734732802*^9, 3.714306738819399*^9}, {3.714306872540366*^9,
|
|
3.714306920827626*^9}},ExpressionUUID->"8d01b7f4-1b47-4870-bd56-\
|
|
c3d01a3b65f0"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0f", "[",
|
|
RowBox[{"4", ",", "0.1", ",", "1", ",", "3"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.71430647467421*^9, 3.71430647483814*^9},
|
|
3.714306561660407*^9, {3.714306701599392*^9, 3.7143067030037622`*^9},
|
|
3.7143068783028393`*^9, {3.714376564639735*^9,
|
|
3.714376573666842*^9}},ExpressionUUID->"ef81724d-7599-4fc6-bc9f-\
|
|
faa6548f8c62"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "2.3012995221671773`*^-6"}], "+",
|
|
RowBox[{"5.475415752242974`*^-6", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714306532722279*^9, {3.714306715505674*^9, 3.714306740900689*^9}, {
|
|
3.714306911904702*^9,
|
|
3.714306922534258*^9}},ExpressionUUID->"97867d61-9c01-430e-8d6c-\
|
|
36bed05cf8e4"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(", "x", ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "s"}], "+", "1"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Sqrt", "[", "\[Pi]", "]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], ")"}],
|
|
RowBox[{"s", "!"}]}], ")"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]], "Input",
|
|
CellChangeTimes->{
|
|
3.714306848086981*^9},ExpressionUUID->"5f8eaeab-440b-4126-be6a-\
|
|
ecb75d7dc208"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"2", " ",
|
|
RowBox[{"ArcSinh", "[", "x", "]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714306848733506*^9},ExpressionUUID->"a05d118b-1aed-4f6a-99b1-\
|
|
0d4033f8afb2"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"x", "^",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2", "s"}], "+", "1"}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], "]"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Sqrt", "[", "\[Pi]", "]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"1", "/", "2"}], "+", "s"}], ")"}],
|
|
RowBox[{"s", "!"}]}], ")"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.714306359943676*^9,
|
|
3.714306410548779*^9}},ExpressionUUID->"ebafb28b-1247-44b5-b0f7-\
|
|
b0ceb9aa700d"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"2", " ",
|
|
RowBox[{"ArcSinh", "[", "x", "]"}]}]], "Output",
|
|
CellChangeTimes->{{3.714306340973686*^9,
|
|
3.7143064115016193`*^9}},ExpressionUUID->"be3bbbd4-9c45-4351-a1ef-\
|
|
63e78194d774"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsSq2n0f", "[",
|
|
RowBox[{"2", ",", "0.1", ",", "1", ",", "1.5"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.714376580704068*^9, 3.714376581601132*^9},
|
|
3.714376625541174*^9},ExpressionUUID->"6ed44f78-9dea-4e9b-9c6f-\
|
|
054174e7e82b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"0.0022397494137522678`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"0.006513830114918795`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{{3.714376584263612*^9, 3.714376626111169*^9},
|
|
3.71437672167714*^9},ExpressionUUID->"7b8e4d31-ecf7-4029-83ce-\
|
|
7d91df23700d"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"NInt", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"k0", " ", "x"}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"-", "q"}], ")"}]}], " ",
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{"I", " ", "k0", " ", "x"}], "]"}], " ", "x", " ",
|
|
RowBox[{"BesselJ", "[",
|
|
RowBox[{"n", ",", " ",
|
|
RowBox[{"k", " ", "x"}]}], "]"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-", "c"}], " ", "x"}], "]"}]}], ")"}], "^", "\[Kappa]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}]], "Input",
|
|
CellChangeTimes->{{3.714376655892346*^9, 3.714376704247987*^9}, {
|
|
3.714376741559724*^9,
|
|
3.7143767467815313`*^9}},ExpressionUUID->"c67c7bd2-b642-48af-8f40-\
|
|
996a591698e8"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"NInt", "[",
|
|
RowBox[{"2", ",", "0", ",", "2", ",", "0.1", ",", "1", ",", "1.5"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714376729806518*^9,
|
|
3.7143767633704863`*^9}},ExpressionUUID->"614cd318-0372-4398-a331-\
|
|
06e8a19a7a2b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"0.0022397494137648237`", "\[VeryThinSpace]", "-",
|
|
RowBox[{"0.006513830114925522`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714376763959959*^9},ExpressionUUID->"88930c3e-c93e-4369-838f-\
|
|
286f7ce31f12"]
|
|
}, Open ]],
|
|
|
|
Cell["Small k, q +n <= 2", "Input",
|
|
CellChangeTimes->{{3.7144498543190403`*^9, 3.7144498940773487`*^9}},
|
|
FormatType->
|
|
"TextForm",ExpressionUUID->"bb88952d-82d8-40c6-9199-8918ec79cfd3"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsShortPoly2", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"2", "^",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "q"}], ")"}]}], "/",
|
|
RowBox[{"Sqrt", "[", "Pi", "]"}]}],
|
|
RowBox[{"Sum", "[", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"k", "^", "n"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n"}], "]"}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "]"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "s"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",", "s"}],
|
|
"]"}],
|
|
RowBox[{
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], ",",
|
|
"s"}], "]"}], "/",
|
|
RowBox[{"Pochhammer", "[",
|
|
RowBox[{
|
|
RowBox[{"1", "+", "n"}], ",", "s"}], "]"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"k", "^", "2"}]}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
"s"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "-", "n"}], ")"}], "/", "2"}]}], "}"}]}],
|
|
"]"}]}], "\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714449900855444*^9, 3.7144500244598913`*^9}, {
|
|
3.714450128012027*^9, 3.714450333156262*^9}, {3.714450377234408*^9,
|
|
3.714450397809031*^9}},ExpressionUUID->"e172db1f-c3c1-45a5-8f3d-\
|
|
170a7c1d9a77"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"NInt", "[",
|
|
RowBox[{"4", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "0.3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.7144503464873953`*^9, 3.7144503490025454`*^9}, {
|
|
3.714450463247095*^9, 3.7144504716236067`*^9},
|
|
3.71445099306275*^9},ExpressionUUID->"01a283df-a1fc-43b5-8a97-\
|
|
7bb7e4ef9fbe"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "0.00009725705172330959`"}], "+",
|
|
RowBox[{"0.0000461397932493922`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714450349678749*^9, 3.71445039986479*^9, {3.714450465787138*^9,
|
|
3.714450473110091*^9}},ExpressionUUID->"7b3fee15-7f5e-46f7-b9ab-\
|
|
64e2cd0009d8"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsShortPoly2", "[",
|
|
RowBox[{"4", ",", "0", ",", "4", ",", "0.1", ",", "1", ",", "1.5"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714450354720892*^9, 3.7144503587185497`*^9}, {
|
|
3.714450476468595*^9,
|
|
3.714450482001546*^9}},ExpressionUUID->"4ee267c3-1395-4600-9e0d-\
|
|
9bef997a3c6c"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"0.`", "\[VeryThinSpace]", "+",
|
|
RowBox[{"0.`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.714450359651774*^9, 3.714450400523344*^9,
|
|
3.714450482387801*^9},ExpressionUUID->"33105fcb-ed74-4431-9f6c-\
|
|
898062630460"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"HsShort", "[",
|
|
RowBox[{
|
|
"q_", ",", "n_", ",", "\[Kappa]_", ",", "c_", ",", "k0_", ",", "k_"}],
|
|
"]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"2", "^",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "q"}], ")"}]}], "/",
|
|
RowBox[{"Sqrt", "[", "Pi", "]"}]}],
|
|
RowBox[{"Sum", "[", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"-", "1"}], ")"}], "^", "\[Sigma]"}], " ",
|
|
RowBox[{"Binomial", "[",
|
|
RowBox[{"\[Kappa]", ",", "\[Sigma]"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"k", "^", "n"}], "/",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"k0", "^", "q"}], " ",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^",
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}]}]}], ")"}]}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"Sum", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"2", "-", "q", "+", "n"}], ")"}], "/", "2"}], "+", "s"}],
|
|
"]"}], " ",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-", "q", "+", "n"}], ")"}], "/", "2"}], "+",
|
|
"s"}], "]"}], "/",
|
|
RowBox[{"Gamma", "[",
|
|
RowBox[{"1", "+", "n", "+", "s"}], "]"}]}], "/",
|
|
RowBox[{"s", "!"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{"k", "^", "2"}]}], "/",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Sigma]", " ", "c"}], "-",
|
|
RowBox[{"I", " ", "k0"}]}], ")"}], "^", "2"}]}], ")"}], "^",
|
|
"s"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"s", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}],
|
|
"\[IndentingNewLine]", ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Sigma]", ",", "0", ",", "\[Kappa]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
CellChangeTimes->{{3.714450806482028*^9, 3.714450835238635*^9}, {
|
|
3.7144509062431183`*^9, 3.71445098397077*^9}, {3.714451423639358*^9,
|
|
3.7144514249172487`*^9}},ExpressionUUID->"f9b1d3e1-f9e3-47df-b296-\
|
|
06925e64f0cf"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"HsShort", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.8", ",", "1", ",", "0.3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714450997511832*^9, 3.7144510807692423`*^9}, {
|
|
3.714451517185413*^9,
|
|
3.714451521313496*^9}},ExpressionUUID->"deb85449-1242-4878-b97a-\
|
|
2c182c410f03"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "0.20418154673151268`"}], "+",
|
|
RowBox[{"0.09043780287300669`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{
|
|
3.714451521584096*^9},ExpressionUUID->"ecbc543a-2554-49e6-9782-\
|
|
90f2af4e63f3"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"NInt", "[",
|
|
RowBox[{"2.001", ",", "0", ",", "4", ",", "0.8", ",", "1", ",", "0.3"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.714451442123979*^9, 3.7144514465635147`*^9}, {
|
|
3.714451527937179*^9,
|
|
3.714451533949725*^9}},ExpressionUUID->"b9f2e7bd-03c7-40fd-904e-\
|
|
770f888fa55e"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"-", "0.20418154496208551`"}], "+",
|
|
RowBox[{"0.09043780575607874`", " ", "\[ImaginaryI]"}]}]], "Output",
|
|
CellChangeTimes->{3.714451449039239*^9,
|
|
3.714451535016184*^9},ExpressionUUID->"fe23d9f8-5eb2-48dd-9916-\
|
|
622dd20f3cdb"]
|
|
}, Open ]]
|
|
},
|
|
Evaluator->"New Kernel 3",
|
|
WindowMargins->{{Automatic, 173}, {187, Automatic}},
|
|
FrontEndVersion->"11.1 for Linux x86 (64-bit) (April 18, 2017)",
|
|
StyleDefinitions->"Default.nb"
|
|
]
|
|
(* End of Notebook Content *)
|
|
|
|
(* Internal cache information *)
|
|
(*CellTagsOutline
|
|
CellTagsIndex->{}
|
|
*)
|
|
(*CellTagsIndex
|
|
CellTagsIndex->{}
|
|
*)
|
|
(*NotebookFileOutline
|
|
Notebook[{
|
|
Cell[558, 20, 1849, 53, 103, "Input", "ExpressionUUID" -> \
|
|
"11f718af-721d-4530-8d00-fc136b151d2a"],
|
|
Cell[CellGroupData[{
|
|
Cell[2432, 77, 1236, 35, 149, "Input", "ExpressionUUID" -> \
|
|
"60b62416-1732-4228-8f52-4dd7ea7c260b"],
|
|
Cell[3671, 114, 288, 5, 34, "Output", "ExpressionUUID" -> \
|
|
"e1c0326f-b1f0-4ebb-9de3-0d92c1b9201a"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[3996, 124, 471, 11, 34, "Input", "ExpressionUUID" -> \
|
|
"9bf951ee-4aec-4260-9ddf-ddd1330b6243"],
|
|
Cell[4470, 137, 3564, 97, 287, "Output", "ExpressionUUID" -> \
|
|
"d2edfa46-cbb3-4c0a-a2db-5b706afe5c8f"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[8071, 239, 402, 8, 34, "Input", "ExpressionUUID" -> \
|
|
"8a8a5ec5-4164-415c-a832-378df85c3809"],
|
|
Cell[8476, 249, 412, 8, 34, "Output", "ExpressionUUID" -> \
|
|
"e6a4d50f-a64a-4c92-bccd-4fd642d8ab43"]
|
|
}, Open ]],
|
|
Cell[8903, 260, 4564, 131, 258, "Input", "ExpressionUUID" -> \
|
|
"15978efa-9e1d-46b2-b638-930987091033"],
|
|
Cell[CellGroupData[{
|
|
Cell[13492, 395, 307, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"bd9bb75c-aa2f-4771-820c-573b5d15abe3"],
|
|
Cell[13802, 404, 293, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"fdc37902-4c0f-48d5-b1f9-b05ced69d16b"]
|
|
}, Open ]],
|
|
Cell[14110, 413, 4678, 133, 236, "Input", "ExpressionUUID" -> \
|
|
"0270ff41-5dea-48e1-8e61-3ad59c8f3c77"],
|
|
Cell[CellGroupData[{
|
|
Cell[18813, 550, 329, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"dc1bf119-f620-4298-9861-0bebd50af39b"],
|
|
Cell[19145, 559, 346, 8, 34, "Output", "ExpressionUUID" -> \
|
|
"502ca379-4501-4726-b8a3-a6d4743d21cf"]
|
|
}, Open ]],
|
|
Cell[19506, 570, 2609, 77, 148, "Input", "ExpressionUUID" -> \
|
|
"7bdc1e3b-21a7-4b53-8781-12296dcb7e43"],
|
|
Cell[CellGroupData[{
|
|
Cell[22140, 651, 254, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"e99ec6cf-d99f-42bb-b94e-4c6feeaefe2f"],
|
|
Cell[22397, 659, 251, 5, 32, "Output", "ExpressionUUID" -> \
|
|
"9defb7fc-3a2a-4372-b667-e589fcb6d801"]
|
|
}, Open ]],
|
|
Cell[22663, 667, 1813, 54, 125, "Input", "ExpressionUUID" -> \
|
|
"c903db13-544e-45bc-b87f-4a3b87557ce7"],
|
|
Cell[CellGroupData[{
|
|
Cell[24501, 725, 254, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"c75220ed-031f-49e5-9467-d46ac941cb11"],
|
|
Cell[24758, 733, 278, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"2ca46fac-dce2-4111-be04-222479370c7c"]
|
|
}, Open ]],
|
|
Cell[25051, 742, 2762, 80, 125, "Input", "ExpressionUUID" -> \
|
|
"91fb5021-708e-4953-92bf-e65236c1db92"],
|
|
Cell[CellGroupData[{
|
|
Cell[27838, 826, 252, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"1d288674-7aa8-40c7-ad24-caf62a39a091"],
|
|
Cell[28093, 834, 316, 7, 32, "Output", "ExpressionUUID" -> \
|
|
"d5266c94-5c79-4a80-a294-09b967a31d30"]
|
|
}, Open ]],
|
|
Cell[28424, 844, 3937, 103, 258, "Input", "ExpressionUUID" -> \
|
|
"35e0146b-a88e-4958-9a81-4af639dd5abb"],
|
|
Cell[CellGroupData[{
|
|
Cell[32386, 951, 301, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"ee4e143e-b867-4930-86b6-de913836ba38"],
|
|
Cell[32690, 960, 283, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"e7215961-ce53-448f-a9e3-0eda8cc30a4d"]
|
|
}, Open ]],
|
|
Cell[32988, 969, 4903, 136, 236, "Input", "ExpressionUUID" -> \
|
|
"a95bd250-d811-48c2-87a5-6e237d1225d7"],
|
|
Cell[CellGroupData[{
|
|
Cell[37916, 1109, 352, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"2ea68715-88f5-4465-af14-0b3fcd58d31b"],
|
|
Cell[38271, 1118, 553, 10, 34, "Output", "ExpressionUUID" -> \
|
|
"a7f32ea2-344b-47e7-80d7-9ffb74f31a08"]
|
|
}, Open ]],
|
|
Cell[38839, 1131, 5709, 158, 282, "Input", "ExpressionUUID" -> \
|
|
"3d8bffca-d6ac-482a-b1a2-b2e45fd6aca8"],
|
|
Cell[CellGroupData[{
|
|
Cell[44573, 1293, 330, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"1abff5df-c88c-4949-93d7-b00aaf2d034c"],
|
|
Cell[44906, 1302, 341, 7, 34, "Output", "ExpressionUUID" -> \
|
|
"69027225-5785-4f7e-9ff2-0a3cee3e176d"]
|
|
}, Open ]],
|
|
Cell[45262, 1312, 269, 6, 32, "Input", "ExpressionUUID" -> \
|
|
"d5926fc0-c5a2-4b56-894e-116ee99824aa"],
|
|
Cell[45534, 1320, 6000, 162, 347, "Input", "ExpressionUUID" -> \
|
|
"5e62d660-2944-47ca-8ee0-e6a553ed7d63"],
|
|
Cell[CellGroupData[{
|
|
Cell[51559, 1486, 354, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"ac03b840-c27b-4fe3-bb0b-82daba30770b"],
|
|
Cell[51916, 1495, 296, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"e5d1f184-d264-4d0a-af52-34362b04d2b6"]
|
|
}, Open ]],
|
|
Cell[52227, 1504, 5350, 141, 259, "Input", "ExpressionUUID" -> \
|
|
"4ab261b6-0c49-491c-9869-75df5adc999d"],
|
|
Cell[CellGroupData[{
|
|
Cell[57602, 1649, 280, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"28a7cebd-2387-4639-b462-54326a81706f"],
|
|
Cell[57885, 1657, 249, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"408303d5-6eab-465e-b43c-43bc1a3b6810"]
|
|
}, Open ]],
|
|
Cell[58149, 1666, 5096, 134, 258, "Input", "ExpressionUUID" -> \
|
|
"a0aeba62-9dcc-4486-a26f-5ef44b8b7a21"],
|
|
Cell[CellGroupData[{
|
|
Cell[63270, 1804, 257, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"959e5694-67a4-4fb4-a01a-51c9949a29f0"],
|
|
Cell[63530, 1812, 247, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"b6a40e5c-36ec-408f-a622-54079303243a"]
|
|
}, Open ]],
|
|
Cell[63792, 1821, 5074, 130, 236, "Input", "ExpressionUUID" -> \
|
|
"811cd081-362a-41a3-bd5d-f77c96ca788c"],
|
|
Cell[CellGroupData[{
|
|
Cell[68891, 1955, 254, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"dc6edbc3-0394-4cd5-be4f-23c469fcc0bb"],
|
|
Cell[69148, 1963, 271, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"d1aa622a-f7c6-4b70-a80c-c09426523c9d"]
|
|
}, Open ]],
|
|
Cell[69434, 1972, 4740, 127, 236, "Input", "ExpressionUUID" -> \
|
|
"747e5737-fbc4-443f-8253-54d4a139ee63"],
|
|
Cell[CellGroupData[{
|
|
Cell[74199, 2103, 283, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"69f1a251-919e-4400-b0c2-e2d5db4dccff"],
|
|
Cell[74485, 2111, 368, 7, 34, "Output", "ExpressionUUID" -> \
|
|
"8b2e58e7-5d5a-4697-824a-0a411cb068bf"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[74890, 2123, 768, 23, 34, "Input", "ExpressionUUID" -> \
|
|
"c015003f-0699-49e2-b866-714eb92d5c6f"],
|
|
Cell[75661, 2148, 311, 10, 63, "Output", "ExpressionUUID" -> \
|
|
"b5efa070-aee7-4251-898d-4ad75f8ee655"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[76009, 2163, 211, 5, 32, "Input", "ExpressionUUID" -> \
|
|
"65bc3149-1111-4cfb-a506-bc566f00020a"],
|
|
Cell[76223, 2170, 147, 4, 35, "Output", "ExpressionUUID" -> \
|
|
"385fc41d-508e-4bcc-87ce-366fda2afa16"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[76407, 2179, 380, 10, 34, "Input", "ExpressionUUID" -> \
|
|
"fbdde755-24a3-46b6-b157-f28e6b6a8022"],
|
|
Cell[76790, 2191, 4401, 90, 230, "Output", "ExpressionUUID" -> \
|
|
"f3bbc0c5-f6fc-4fd8-ade3-1015c944f07e"]
|
|
}, Open ]],
|
|
Cell[81206, 2284, 3362, 93, 191, "Input", "ExpressionUUID" -> \
|
|
"e139a925-f883-4746-b86a-c2602253404c"],
|
|
Cell[CellGroupData[{
|
|
Cell[84593, 2381, 336, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"7c490700-b77f-4e41-bc26-5314e7de681a"],
|
|
Cell[84932, 2390, 345, 8, 34, "Output", "ExpressionUUID" -> \
|
|
"2ec8c9bc-264b-4c99-966f-0e16399c5a47"]
|
|
}, Open ]],
|
|
Cell[85292, 2401, 3279, 88, 191, "Input", "ExpressionUUID" -> \
|
|
"03b0ced6-4edd-459d-b511-4346ebf0b6b3"],
|
|
Cell[CellGroupData[{
|
|
Cell[88596, 2493, 334, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"56fbb4e1-8f74-4c7e-a055-064f55e0a5e4"],
|
|
Cell[88933, 2502, 419, 9, 34, "Output", "ExpressionUUID" -> \
|
|
"ccc74805-25fc-4b7a-92ea-ecc47f1289c3"]
|
|
}, Open ]],
|
|
Cell[89367, 2514, 1338, 41, 79, "Input", "ExpressionUUID" -> \
|
|
"fdc5165f-652c-4553-b7d6-e673dd5aaf0c"],
|
|
Cell[CellGroupData[{
|
|
Cell[90730, 2559, 259, 5, 34, "Input", "ExpressionUUID" -> \
|
|
"14bbf919-b4de-4582-b295-f820ec4e1863"],
|
|
Cell[90992, 2566, 233, 5, 34, "Output", "ExpressionUUID" -> \
|
|
"6f771c74-3b2b-4814-8148-354f0c429799"]
|
|
}, Open ]],
|
|
Cell[91240, 2574, 2075, 58, 103, "Input", "ExpressionUUID" -> \
|
|
"4fe8906e-3924-4127-922e-cba66e63b281"],
|
|
Cell[CellGroupData[{
|
|
Cell[93340, 2636, 283, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"314b1bc8-d2f0-41e7-a76a-821f0a313b1d"],
|
|
Cell[93626, 2644, 394, 8, 34, "Output", "ExpressionUUID" -> \
|
|
"a7a1768d-8381-468e-9182-cad6264fbeb4"]
|
|
}, Open ]],
|
|
Cell[94035, 2655, 1830, 55, 103, "Input", "ExpressionUUID" -> \
|
|
"1b478334-5bf6-4ef1-a489-1d672116aa7a"],
|
|
Cell[CellGroupData[{
|
|
Cell[95890, 2714, 230, 5, 34, "Input", "ExpressionUUID" -> \
|
|
"e3c36c04-d161-4900-98a4-33e7dc51e872"],
|
|
Cell[96123, 2721, 248, 6, 34, "Output", "ExpressionUUID" -> \
|
|
"fed7bff7-9bbc-40e2-8c1c-b16c41ea5128"]
|
|
}, Open ]],
|
|
Cell[96386, 2730, 1826, 53, 103, "Input", "ExpressionUUID" -> \
|
|
"5d9b2d82-846d-4549-b840-40cefa18d8c0"],
|
|
Cell[CellGroupData[{
|
|
Cell[98237, 2787, 305, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"d9d5283f-d204-4f19-83d0-7773fcf3cd30"],
|
|
Cell[98545, 2795, 298, 7, 34, "Output", "ExpressionUUID" -> \
|
|
"09894aad-397c-465b-b0aa-c9f601a9aec6"]
|
|
}, Open ]],
|
|
Cell[98858, 2805, 1048, 28, 57, "Input", "ExpressionUUID" -> \
|
|
"8d01b7f4-1b47-4870-bd56-c3d01a3b65f0"],
|
|
Cell[CellGroupData[{
|
|
Cell[99931, 2837, 379, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"ef81724d-7599-4fc6-bc9f-faa6548f8c62"],
|
|
Cell[100313, 2846, 348, 8, 34, "Output", "ExpressionUUID" -> \
|
|
"97867d61-9c01-430e-8d6c-36bed05cf8e4"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[100698, 2859, 797, 27, 34, "Input", "ExpressionUUID" -> \
|
|
"5f8eaeab-440b-4126-be6a-ecb75d7dc208"],
|
|
Cell[101498, 2888, 186, 5, 32, "Output", "ExpressionUUID" -> \
|
|
"a05d118b-1aed-4f6a-99b1-0d4033f8afb2"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[101721, 2898, 795, 26, 34, "Input", "ExpressionUUID" -> \
|
|
"ebafb28b-1247-44b5-b0f7-b0ceb9aa700d"],
|
|
Cell[102519, 2926, 212, 5, 32, "Output", "ExpressionUUID" -> \
|
|
"be3bbbd4-9c45-4351-a1ef-63e78194d774"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[102768, 2936, 257, 5, 34, "Input", "ExpressionUUID" -> \
|
|
"6ed44f78-9dea-4e9b-9c6f-054174e7e82b"],
|
|
Cell[103028, 2943, 294, 5, 32, "Output", "ExpressionUUID" -> \
|
|
"7b8e4d31-ecf7-4029-83ce-7d91df23700d"]
|
|
}, Open ]],
|
|
Cell[103337, 2951, 1016, 31, 57, "Input", "ExpressionUUID" -> \
|
|
"c67c7bd2-b642-48af-8f40-996a591698e8"],
|
|
Cell[CellGroupData[{
|
|
Cell[104378, 2986, 255, 6, 34, "Input", "ExpressionUUID" -> \
|
|
"614cd318-0372-4398-a331-06e8a19a7a2b"],
|
|
Cell[104636, 2994, 248, 5, 32, "Output", "ExpressionUUID" -> \
|
|
"88930c3e-c93e-4369-838f-286f7ce31f12"]
|
|
}, Open ]],
|
|
Cell[104899, 3002, 188, 3, 32, "Input", "ExpressionUUID" -> \
|
|
"bb88952d-82d8-40c6-9199-8918ec79cfd3"],
|
|
Cell[105090, 3007, 3046, 91, 171, "Input", "ExpressionUUID" -> \
|
|
"e172db1f-c3c1-45a5-8f3d-170a7c1d9a77"],
|
|
Cell[CellGroupData[{
|
|
Cell[108161, 3102, 331, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"01a283df-a1fc-43b5-8a97-7bb7e4ef9fbe"],
|
|
Cell[108495, 3111, 319, 7, 32, "Output", "ExpressionUUID" -> \
|
|
"7b3fee15-7f5e-46f7-b9ab-64e2cd0009d8"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[108851, 3123, 312, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"4ee267c3-1395-4600-9e0d-9bef997a3c6c"],
|
|
Cell[109166, 3132, 255, 5, 32, "Output", "ExpressionUUID" -> \
|
|
"33105fcb-ed74-4431-9f6c-898062630460"]
|
|
}, Open ]],
|
|
Cell[109436, 3140, 2464, 73, 125, "Input", "ExpressionUUID" -> \
|
|
"f9b1d3e1-f9e3-47df-b296-06925e64f0cf"],
|
|
Cell[CellGroupData[{
|
|
Cell[111925, 3217, 311, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"deb85449-1242-4878-b97a-2c182c410f03"],
|
|
Cell[112239, 3226, 243, 6, 32, "Output", "ExpressionUUID" -> \
|
|
"ecbc543a-2554-49e6-9782-90f2af4e63f3"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[112519, 3237, 308, 7, 34, "Input", "ExpressionUUID" -> \
|
|
"b9f2e7bd-03c7-40fd-904e-770f888fa55e"],
|
|
Cell[112830, 3246, 265, 6, 32, "Output", "ExpressionUUID" -> \
|
|
"fe23d9f8-5eb2-48dd-9916-622dd20f3cdb"]
|
|
}, Open ]]
|
|
}
|
|
]
|
|
*)
|
|
|