diff --git a/notes/BesselTFnumtests.nb b/notes/BesselTFnumtests.nb new file mode 100644 index 0000000..f9b5ea8 --- /dev/null +++ b/notes/BesselTFnumtests.nb @@ -0,0 +1,3166 @@ +(* 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[ 111859, 3158] +NotebookOptionsPosition[ 102675, 2933] +NotebookOutlinePosition[ 103042, 2949] +CellTagsIndexPosition[ 102999, 2946] +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},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},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 ]] +}, +Evaluator->"New Kernel 3", +WindowSize->{Full, Full}, +WindowMargins->{{Automatic, -244}, {Automatic, 405}}, +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, 34, "Input", "ExpressionUUID" -> \ +"11f718af-721d-4530-8d00-fc136b151d2a"], +Cell[CellGroupData[{ +Cell[2432, 77, 1214, 35, 126, "Input", "ExpressionUUID" -> \ +"60b62416-1732-4228-8f52-4dd7ea7c260b"], +Cell[3649, 114, 288, 5, 34, "Output", "ExpressionUUID" -> \ +"e1c0326f-b1f0-4ebb-9de3-0d92c1b9201a"] +}, Open ]], +Cell[CellGroupData[{ +Cell[3974, 124, 471, 11, 34, "Input", "ExpressionUUID" -> \ +"9bf951ee-4aec-4260-9ddf-ddd1330b6243"], +Cell[4448, 137, 3564, 97, 133, "Output", "ExpressionUUID" -> \ +"d2edfa46-cbb3-4c0a-a2db-5b706afe5c8f"] +}, Open ]], +Cell[CellGroupData[{ +Cell[8049, 239, 402, 8, 34, "Input", "ExpressionUUID" -> \ +"8a8a5ec5-4164-415c-a832-378df85c3809"], +Cell[8454, 249, 412, 8, 34, "Output", "ExpressionUUID" -> \ +"e6a4d50f-a64a-4c92-bccd-4fd642d8ab43"] +}, Open ]], +Cell[8881, 260, 4564, 131, 147, "Input", "ExpressionUUID" -> \ +"15978efa-9e1d-46b2-b638-930987091033"], +Cell[CellGroupData[{ +Cell[13470, 395, 307, 7, 34, "Input", "ExpressionUUID" -> \ +"bd9bb75c-aa2f-4771-820c-573b5d15abe3"], +Cell[13780, 404, 293, 6, 34, "Output", "ExpressionUUID" -> \ +"fdc37902-4c0f-48d5-b1f9-b05ced69d16b"] +}, Open ]], +Cell[14088, 413, 4678, 133, 147, "Input", "ExpressionUUID" -> \ +"0270ff41-5dea-48e1-8e61-3ad59c8f3c77"], +Cell[CellGroupData[{ +Cell[18791, 550, 329, 7, 34, "Input", "ExpressionUUID" -> \ +"dc1bf119-f620-4298-9861-0bebd50af39b"], +Cell[19123, 559, 346, 8, 34, "Output", "ExpressionUUID" -> \ +"502ca379-4501-4726-b8a3-a6d4743d21cf"] +}, Open ]], +Cell[19484, 570, 2609, 77, 125, "Input", "ExpressionUUID" -> \ +"7bdc1e3b-21a7-4b53-8781-12296dcb7e43"], +Cell[CellGroupData[{ +Cell[22118, 651, 254, 6, 34, "Input", "ExpressionUUID" -> \ +"e99ec6cf-d99f-42bb-b94e-4c6feeaefe2f"], +Cell[22375, 659, 251, 5, 32, "Output", "ExpressionUUID" -> \ +"9defb7fc-3a2a-4372-b667-e589fcb6d801"] +}, Open ]], +Cell[22641, 667, 1813, 54, 34, "Input", "ExpressionUUID" -> \ +"c903db13-544e-45bc-b87f-4a3b87557ce7"], +Cell[CellGroupData[{ +Cell[24479, 725, 254, 6, 34, "Input", "ExpressionUUID" -> \ +"c75220ed-031f-49e5-9467-d46ac941cb11"], +Cell[24736, 733, 278, 6, 34, "Output", "ExpressionUUID" -> \ +"2ca46fac-dce2-4111-be04-222479370c7c"] +}, Open ]], +Cell[25029, 742, 2762, 80, 125, "Input", "ExpressionUUID" -> \ +"91fb5021-708e-4953-92bf-e65236c1db92"], +Cell[CellGroupData[{ +Cell[27816, 826, 252, 6, 34, "Input", "ExpressionUUID" -> \ +"1d288674-7aa8-40c7-ad24-caf62a39a091"], +Cell[28071, 834, 316, 7, 32, "Output", "ExpressionUUID" -> \ +"d5266c94-5c79-4a80-a294-09b967a31d30"] +}, Open ]], +Cell[28402, 844, 3937, 103, 235, "Input", "ExpressionUUID" -> \ +"35e0146b-a88e-4958-9a81-4af639dd5abb"], +Cell[CellGroupData[{ +Cell[32364, 951, 301, 7, 34, "Input", "ExpressionUUID" -> \ +"ee4e143e-b867-4930-86b6-de913836ba38"], +Cell[32668, 960, 283, 6, 34, "Output", "ExpressionUUID" -> \ +"e7215961-ce53-448f-a9e3-0eda8cc30a4d"] +}, Open ]], +Cell[32966, 969, 4903, 136, 170, "Input", "ExpressionUUID" -> \ +"a95bd250-d811-48c2-87a5-6e237d1225d7"], +Cell[CellGroupData[{ +Cell[37894, 1109, 352, 7, 34, "Input", "ExpressionUUID" -> \ +"2ea68715-88f5-4465-af14-0b3fcd58d31b"], +Cell[38249, 1118, 553, 10, 34, "Output", "ExpressionUUID" -> \ +"a7f32ea2-344b-47e7-80d7-9ffb74f31a08"] +}, Open ]], +Cell[38817, 1131, 5709, 158, 238, "Input", "ExpressionUUID" -> \ +"3d8bffca-d6ac-482a-b1a2-b2e45fd6aca8"], +Cell[CellGroupData[{ +Cell[44551, 1293, 330, 7, 34, "Input", "ExpressionUUID" -> \ +"1abff5df-c88c-4949-93d7-b00aaf2d034c"], +Cell[44884, 1302, 341, 7, 34, "Output", "ExpressionUUID" -> \ +"69027225-5785-4f7e-9ff2-0a3cee3e176d"] +}, Open ]], +Cell[45240, 1312, 269, 6, 32, "Input", "ExpressionUUID" -> \ +"d5926fc0-c5a2-4b56-894e-116ee99824aa"], +Cell[45512, 1320, 6000, 162, 259, "Input", "ExpressionUUID" -> \ +"5e62d660-2944-47ca-8ee0-e6a553ed7d63"], +Cell[CellGroupData[{ +Cell[51537, 1486, 354, 7, 34, "Input", "ExpressionUUID" -> \ +"ac03b840-c27b-4fe3-bb0b-82daba30770b"], +Cell[51894, 1495, 296, 6, 34, "Output", "ExpressionUUID" -> \ +"e5d1f184-d264-4d0a-af52-34362b04d2b6"] +}, Open ]], +Cell[52205, 1504, 5350, 141, 237, "Input", "ExpressionUUID" -> \ +"4ab261b6-0c49-491c-9869-75df5adc999d"], +Cell[CellGroupData[{ +Cell[57580, 1649, 280, 6, 34, "Input", "ExpressionUUID" -> \ +"28a7cebd-2387-4639-b462-54326a81706f"], +Cell[57863, 1657, 249, 6, 34, "Output", "ExpressionUUID" -> \ +"408303d5-6eab-465e-b43c-43bc1a3b6810"] +}, Open ]], +Cell[58127, 1666, 5096, 134, 258, "Input", "ExpressionUUID" -> \ +"a0aeba62-9dcc-4486-a26f-5ef44b8b7a21"], +Cell[CellGroupData[{ +Cell[63248, 1804, 257, 6, 34, "Input", "ExpressionUUID" -> \ +"959e5694-67a4-4fb4-a01a-51c9949a29f0"], +Cell[63508, 1812, 247, 6, 34, "Output", "ExpressionUUID" -> \ +"b6a40e5c-36ec-408f-a622-54079303243a"] +}, Open ]], +Cell[63770, 1821, 5074, 130, 236, "Input", "ExpressionUUID" -> \ +"811cd081-362a-41a3-bd5d-f77c96ca788c"], +Cell[CellGroupData[{ +Cell[68869, 1955, 254, 6, 34, "Input", "ExpressionUUID" -> \ +"dc6edbc3-0394-4cd5-be4f-23c469fcc0bb"], +Cell[69126, 1963, 271, 6, 34, "Output", "ExpressionUUID" -> \ +"d1aa622a-f7c6-4b70-a80c-c09426523c9d"] +}, Open ]], +Cell[69412, 1972, 4740, 127, 236, "Input", "ExpressionUUID" -> \ +"747e5737-fbc4-443f-8253-54d4a139ee63"], +Cell[CellGroupData[{ +Cell[74177, 2103, 283, 6, 34, "Input", "ExpressionUUID" -> \ +"69f1a251-919e-4400-b0c2-e2d5db4dccff"], +Cell[74463, 2111, 368, 7, 34, "Output", "ExpressionUUID" -> \ +"8b2e58e7-5d5a-4697-824a-0a411cb068bf"] +}, Open ]], +Cell[CellGroupData[{ +Cell[74868, 2123, 768, 23, 34, "Input", "ExpressionUUID" -> \ +"c015003f-0699-49e2-b866-714eb92d5c6f"], +Cell[75639, 2148, 311, 10, 63, "Output", "ExpressionUUID" -> \ +"b5efa070-aee7-4251-898d-4ad75f8ee655"] +}, Open ]], +Cell[CellGroupData[{ +Cell[75987, 2163, 211, 5, 32, "Input", "ExpressionUUID" -> \ +"65bc3149-1111-4cfb-a506-bc566f00020a"], +Cell[76201, 2170, 147, 4, 35, "Output", "ExpressionUUID" -> \ +"385fc41d-508e-4bcc-87ce-366fda2afa16"] +}, Open ]], +Cell[CellGroupData[{ +Cell[76385, 2179, 380, 10, 34, "Input", "ExpressionUUID" -> \ +"fbdde755-24a3-46b6-b157-f28e6b6a8022"], +Cell[76768, 2191, 4401, 90, 230, "Output", "ExpressionUUID" -> \ +"f3bbc0c5-f6fc-4fd8-ade3-1015c944f07e"] +}, Open ]], +Cell[81184, 2284, 3362, 93, 191, "Input", "ExpressionUUID" -> \ +"e139a925-f883-4746-b86a-c2602253404c"], +Cell[CellGroupData[{ +Cell[84571, 2381, 336, 7, 34, "Input", "ExpressionUUID" -> \ +"7c490700-b77f-4e41-bc26-5314e7de681a"], +Cell[84910, 2390, 345, 8, 34, "Output", "ExpressionUUID" -> \ +"2ec8c9bc-264b-4c99-966f-0e16399c5a47"] +}, Open ]], +Cell[85270, 2401, 3279, 88, 191, "Input", "ExpressionUUID" -> \ +"03b0ced6-4edd-459d-b511-4346ebf0b6b3"], +Cell[CellGroupData[{ +Cell[88574, 2493, 334, 7, 34, "Input", "ExpressionUUID" -> \ +"56fbb4e1-8f74-4c7e-a055-064f55e0a5e4"], +Cell[88911, 2502, 419, 9, 34, "Output", "ExpressionUUID" -> \ +"ccc74805-25fc-4b7a-92ea-ecc47f1289c3"] +}, Open ]], +Cell[89345, 2514, 1338, 41, 79, "Input", "ExpressionUUID" -> \ +"fdc5165f-652c-4553-b7d6-e673dd5aaf0c"], +Cell[CellGroupData[{ +Cell[90708, 2559, 259, 5, 34, "Input", "ExpressionUUID" -> \ +"14bbf919-b4de-4582-b295-f820ec4e1863"], +Cell[90970, 2566, 233, 5, 34, "Output", "ExpressionUUID" -> \ +"6f771c74-3b2b-4814-8148-354f0c429799"] +}, Open ]], +Cell[91218, 2574, 2075, 58, 34, "Input", "ExpressionUUID" -> \ +"4fe8906e-3924-4127-922e-cba66e63b281"], +Cell[CellGroupData[{ +Cell[93318, 2636, 283, 6, 34, "Input", "ExpressionUUID" -> \ +"314b1bc8-d2f0-41e7-a76a-821f0a313b1d"], +Cell[93604, 2644, 394, 8, 34, "Output", "ExpressionUUID" -> \ +"a7a1768d-8381-468e-9182-cad6264fbeb4"] +}, Open ]], +Cell[94013, 2655, 1830, 55, 34, "Input", "ExpressionUUID" -> \ +"1b478334-5bf6-4ef1-a489-1d672116aa7a"], +Cell[CellGroupData[{ +Cell[95868, 2714, 230, 5, 34, "Input", "ExpressionUUID" -> \ +"e3c36c04-d161-4900-98a4-33e7dc51e872"], +Cell[96101, 2721, 248, 6, 34, "Output", "ExpressionUUID" -> \ +"fed7bff7-9bbc-40e2-8c1c-b16c41ea5128"] +}, Open ]], +Cell[96364, 2730, 1826, 53, 34, "Input", "ExpressionUUID" -> \ +"5d9b2d82-846d-4549-b840-40cefa18d8c0"], +Cell[CellGroupData[{ +Cell[98215, 2787, 305, 6, 34, "Input", "ExpressionUUID" -> \ +"d9d5283f-d204-4f19-83d0-7773fcf3cd30"], +Cell[98523, 2795, 298, 7, 34, "Output", "ExpressionUUID" -> \ +"09894aad-397c-465b-b0aa-c9f601a9aec6"] +}, Open ]], +Cell[98836, 2805, 1048, 28, 34, "Input", "ExpressionUUID" -> \ +"8d01b7f4-1b47-4870-bd56-c3d01a3b65f0"], +Cell[CellGroupData[{ +Cell[99909, 2837, 329, 6, 34, "Input", "ExpressionUUID" -> \ +"ef81724d-7599-4fc6-bc9f-faa6548f8c62"], +Cell[100241, 2845, 348, 8, 34, "Output", "ExpressionUUID" -> \ +"97867d61-9c01-430e-8d6c-36bed05cf8e4"] +}, Open ]], +Cell[CellGroupData[{ +Cell[100626, 2858, 797, 27, 34, "Input", "ExpressionUUID" -> \ +"5f8eaeab-440b-4126-be6a-ecb75d7dc208"], +Cell[101426, 2887, 186, 5, 32, "Output", "ExpressionUUID" -> \ +"a05d118b-1aed-4f6a-99b1-0d4033f8afb2"] +}, Open ]], +Cell[CellGroupData[{ +Cell[101649, 2897, 795, 26, 34, "Input", "ExpressionUUID" -> \ +"ebafb28b-1247-44b5-b0f7-b0ceb9aa700d"], +Cell[102447, 2925, 212, 5, 32, "Output", "ExpressionUUID" -> \ +"be3bbbd4-9c45-4351-a1ef-63e78194d774"] +}, Open ]] +} +] +*) + diff --git a/notes/ewald-calculations-apr1.lyx b/notes/ewald-calculations-apr1.lyx new file mode 100644 index 0000000..770f59f --- /dev/null +++ b/notes/ewald-calculations-apr1.lyx @@ -0,0 +1,377 @@ +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 +\begin_document +\begin_header +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language finnish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman TeX Gyre Pagella +\font_sans default +\font_typewriter default +\font_math auto +\font_default_family default +\use_non_tex_fonts true +\font_sc false +\font_osf true +\font_sf_scale 100 +\font_tt_scale 100 +\graphics default +\default_output_format pdf4 +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize 10 +\spacing single +\use_hyperref true +\pdf_title "Sähköpajan päiväkirja" +\pdf_author "Marek Nečada" +\pdf_bookmarks true +\pdf_bookmarksnumbered false +\pdf_bookmarksopen false +\pdf_bookmarksopenlevel 1 +\pdf_breaklinks false +\pdf_pdfborder false +\pdf_colorlinks false +\pdf_backref false +\pdf_pdfusetitle true +\papersize a3paper +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 1cm +\topmargin 5mm +\rightmargin 1cm +\bottommargin 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\quotes_language swedish +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard + +\lang english +\begin_inset FormulaMacro +\newcommand{\uoft}[1]{\mathfrak{F}#1} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\usht}[2]{\mathbb{S}_{#1}#2} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\bsht}[2]{\mathrm{S}_{#1}#2} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\vect}[1]{\mathbf{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ud}{\mathrm{d}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\basis}[1]{\mathfrak{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\dc}[1]{Ш_{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\rec}[1]{#1^{-1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\recb}[1]{#1^{\widehat{-1}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ints}{\mathbb{Z}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\nats}{\mathbb{N}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\reals}{\mathbb{R}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ush}[2]{Y_{#1,#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\hgfr}{\mathbf{F}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ph}{\mathrm{ph}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\kor}[1]{\underline{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\koru}[1]{\overline{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\hgf}{F} +\end_inset + +Let +\end_layout + +\begin_layout Paragraph + +\lang english +Large k +\end_layout + +\begin_layout Standard + +\lang english +\begin_inset Formula +\begin{eqnarray*} +\mbox{OK}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\ +\mbox{OK(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\ + & & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{Γ\left(\frac{3-q+n}{2}\right)\text{Γ}\left(1+n-\frac{2-q+n}{2}\right)}\hgfr\left(\begin{array}{c} +\frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\ +1/2 +\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\ + & - & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(1+n-\frac{3-q+n}{2}\right)}\hgfr\left(\begin{array}{c} +\frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\ +3/2 +\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\ +\mbox{OK20} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\ + & & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\hgfr\left(\begin{array}{c} +\frac{2-q+n}{2},\frac{2-q-n}{2}\\ +1/2 +\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\ + & - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\hgfr\left(\begin{array}{c} +\frac{3-q+n}{2},\frac{3-q-n}{2}\\ +3/2 +\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\ +\mbox{(D15.2.2)OK3a,b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\ + & & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{1}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s}\\ + & - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s})\\ +\mbox{OK4a} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\ + & & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}k^{-2+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{2-q+n}+2s}\\ + & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}k^{-3+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{3-q+n}+2s})\\ +\mbox{OK4b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\ + & & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\kor{k^{-2+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{2s}}\\ + & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\kor{k^{-3+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{1+2s}})\\ +\mbox{OK4c} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=\kor 0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\ + & & \times\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ +\mbox{OK4d} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right) +\end{eqnarray*} + +\end_inset + +the fact that the partial sum +\begin_inset Formula $\sum_{s=0}^{\left\lceil \kappa/2\right\rceil -1}\ldots$ +\end_inset + + is zero is shown in the old messy notes (or TODO later here) +\end_layout + +\begin_layout Standard + +\lang english +Using DLMF 5.5.5, which says +\begin_inset Formula $Γ(2z)=\pi^{-1/2}2^{2z-1}\text{Γ}(z)\text{Γ}(z+\frac{1}{2})$ +\end_inset + + we have +\begin_inset Formula +\[ +\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right), +\] + +\end_inset + +so +\begin_inset Formula +\begin{eqnarray*} +\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{\text{Γ}\left(2-q+n\right)}}{\kor{2^{n}}k_{0}^{q}}\kor{\pi}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)}\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{2^{1-q}}}{k_{0}^{q}}\koru{\sqrt{\pi}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\kor{\koru{\text{Γ}\left(\frac{2-q+n}{2}\right)}\left(\frac{2-q+n}{2}\right)_{s}}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\kor{\koru{\text{Γ}\left(\frac{3-q+n}{2}\right)}\left(\frac{3-q+n}{2}\right)_{s}}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right) +\end{eqnarray*} + +\end_inset + +Assuming that +\begin_inset Formula $\left\lceil \frac{\kappa}{2}\right\rceil $ +\end_inset + + is large enough so that all the divergent terms are cancelled, either the + left or the right part will become finite sums due to the +\begin_inset Quotes sld +\end_inset + +extra +\begin_inset Quotes srd +\end_inset + + Pochhammer +\begin_inset Formula $\left(\frac{3-q-n}{2}\right)_{s}$ +\end_inset + + or +\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}$ +\end_inset + +. +\end_layout + +\begin_layout Subparagraph + +\lang english +Special case +\begin_inset Formula $q=2,n=0$ +\end_inset + + +\end_layout + +\begin_layout Standard + +\lang english +If +\begin_inset Formula $\kappa\ge2$ +\end_inset + +, the left part will drop and +\begin_inset Formula +\begin{eqnarray*} +\mbox{OKSq2n0b}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\kor{\text{Γ}\left(\frac{3}{2}+s\right)}s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\kor{\text{Γ}\left(\frac{1}{2}+s\right)}\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\koru{\kor{\text{Γ}\left(\frac{1}{2}+s\right)}\left(\frac{1}{2}+s\right)}s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\kor{\left\lceil \frac{\kappa}{2}\right\rceil }}^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\left(\frac{1}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ +\mbox{(explain!)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\koru 0}^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\kor{\text{Γ}\left(\frac{1}{2}\right)}\left(\frac{1}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ + & = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}\sqrt{\pi}}\frac{\left(\sigma c-ik_{0}\right)}{k}\kor{\sum_{s=0}^{\infty}\left(-1\right)^{s}\left(\frac{\sigma c-ik_{0}}{k}\right)^{2s}\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\left(\frac{1}{2}+s\right)s!}}\\ + & = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}\sqrt{\pi}}\frac{\left(\sigma c-ik_{0}\right)}{k}\frac{2\sqrt{\pi}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)}{\frac{\sigma c-ik_{0}}{k}}\\ +\mbox{OKSq2n0f} & = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{1}{k_{0}^{2}}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right) +\end{eqnarray*} + +\end_inset + +where we used (TODO ref) +\begin_inset Formula +\[ +\sum_{s=0}^{\infty}\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\left(\frac{1}{2}+s\right)s!}\left(-x\right)^{s}=\frac{2\sqrt{\pi}\sinh^{-1}\sqrt{x}}{\sqrt{x}} +\] + +\end_inset + +The final result has asymptotic behaviour of ... + for +\begin_inset Formula $k\to\infty$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Small k +\end_layout + +\begin_layout Standard + +\lang english +\begin_inset Formula +\begin{eqnarray*} +\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\ +\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\kor{Γ\left(2-q+n\right)}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\kor{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{3-q+n}{2}\right)_{s}}}{Γ(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s},\quad\left|\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right|<1 +\end{eqnarray*} + +\end_inset + +Again we use +\begin_inset Formula +\[ +\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right), +\] + +\end_inset + + so +\begin_inset Formula +\begin{eqnarray*} +\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \koru{\frac{2^{1-q+n}}{\sqrt{\pi}}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\text{Γ}\left(\frac{3-q+n}{2}+s\right)}}{\text{Γ}(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s}\\ + & = & \frac{2^{1-q+n}}{\sqrt{\pi}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\text{Γ}\left(\frac{3-q+n}{2}+s\right)}{\text{Γ}(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s} +\end{eqnarray*} + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/notes/ewald-calculations.lyx b/notes/ewald-calculations.lyx index 9611803..21c1612 100644 --- a/notes/ewald-calculations.lyx +++ b/notes/ewald-calculations.lyx @@ -208,8 +208,8 @@ Let \lang english \begin_inset Formula \begin{eqnarray*} -\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\ -\mbox{(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\ +\mbox{OK}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\ +\mbox{OK(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\ & & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{Γ\left(\frac{3-q+n}{2}\right)\text{Γ}\left(1+n-\frac{2-q+n}{2}\right)}\hgfr\left(\begin{array}{c} \frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\ 1/2 @@ -218,7 +218,7 @@ Let \frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\ 3/2 \end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\ - & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\ +\mbox{OK20} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\ & & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\hgfr\left(\begin{array}{c} \frac{2-q+n}{2},\frac{2-q-n}{2}\\ 1/2 @@ -227,16 +227,16 @@ Let \frac{3-q+n}{2},\frac{3-q-n}{2}\\ 3/2 \end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\ -\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\ +\mbox{(D15.2.2)OK3a,b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\ & & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{1}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s}\\ & - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s})\\ - & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\ +\mbox{OK4a} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\ & & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}k^{-2+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{2-q+n}+2s}\\ & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}k^{-3+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{3-q+n}+2s})\\ -\mbox{} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\ +\mbox{OK4b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\ & & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\kor{k^{-2+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{2s}}\\ & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\kor{k^{-3+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{1+2s}})\\ - & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\ +\mbox{OK4c} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\ & & \times\left(\underbrace{\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}}_{\equiv c_{q,n,s}}-\underbrace{\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}}_{č_{q,n,s}}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\kor{\left(\sigma c-ik_{0}\right)^{2s}}c_{q,n,s}-\frac{\left(\sigma c-ik_{0}\right)^{2s+1}}{k}č_{q,n,s}\right)\\ \mbox{(binom.)} & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(c_{q,n,s}\sum_{t=0}^{2s}\binom{2s}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s-t}-č_{q,n,s}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\ @@ -383,13 +383,13 @@ If \begin_inset Formula \begin{eqnarray*} -\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\hgfr}\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\ -\mbox{(D15.1.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)\koru{\text{Γ}(1+n)}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\koru{\hgf}\left(\frac{2-q+n}{2},\kor{\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}\right)\\ -\mbox{(D15.8.6)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\koru{\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}}\hgf\left(\begin{array}{c} +\mbox{OK}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\hgfr}\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\ +\mbox{\ensuremath{\mbox{OK}}(D15.1.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}\koru{\text{Γ}(1+n)}}\koru{\hgf}\left(\frac{2-q+n}{2},\kor{\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}\right)\\ +\mbox{(D15.8.6)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}\text{Γ}(1+n)}\koru{\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}}\hgf\left(\begin{array}{c} \frac{2-q+n}{2},\koru{\kor{1-\left(1+n\right)+\frac{2-q+n}{2}}}\\ \koru{\kor{1-\frac{3-q+n}{2}+\frac{2-q+n}{2}}} \end{array};\koru{\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}}\right)\\ - & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{k^{q-2}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\koru{\frac{3}{2}\left(2-q+n\right)}}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c} +\mbox{NOTOK} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{k^{q-2}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\koru{\frac{3}{2}\left(2-q+n\right)}}\text{Γ}(1+n)}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c} \frac{2-q+n}{2},\koru{\frac{2-q-n}{2}}\\ \koru{1/2} \end{array};\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)}\\ diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 58271c4..7a32112 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -2680,97 +2680,25 @@ Case \end_layout \begin_layout Standard -[REF -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -begin{russian} +As shown in a separate note, \end_layout -\end_inset - -Прудников, том 2 -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{russian} -\end_layout - -\end_inset - -, 2.12.10.2] provides the following integral -\begin_inset Formula -\begin{multline*} -\int_{0}^{\infty}\frac{1}{x^{2}}e^{-px-b/x}J_{0}(cx)\,\ud x=2c\left[z_{+}^{-1}J_{1}\left(z_{-}\right)K_{0}\left(z_{+}\right)+z_{-}^{-1}J_{0}\left(z_{-}\right)K_{1}\left(z_{+}\right)\right]\\ -\left[z_{\pm}=\sqrt{2b}\left(\sqrt{p^{2}+c^{2}}\pm p\right)^{1/2};\Re p>\left|\Im c\right|;\Re b>0\right] -\end{multline*} - -\end_inset - -where [REF -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -begin{russian} -\end_layout - -\end_inset - -Прудников, том 2 -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{russian} -\end_layout - -\end_inset - -, p.659] -\begin_inset Formula $K_{\nu}(z)$ -\end_inset - - is the Macdonald's function (modified Bessel function of 3rd kind) +\begin_layout Standard \begin_inset Formula \[ -K_{\nu}\left(z\right)=\frac{\pi\left[I_{-\nu}\left(z\right)-I_{\nu}\left(z\right)\right]}{2\sin\nu\pi}\quad\left[\nu\notin\ints\right],\quad K_{n}\left(z\right)=\lim_{\nu\to n}K_{\nu}\left(z\right)\quad\left[n\in\ints\right], +\pht 0{s_{2,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=-\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{1}{k_{0}^{2}}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right) \] \end_inset -and -\begin_inset Formula $I_{\nu}\left(z\right)$ +for +\begin_inset Formula $\kappa\ge?$ \end_inset - is the modified Bessel function of 1st kind -\begin_inset Formula -\[ -I_{\nu}\left(z\right)=\frac{1}{\Gamma\left(\nu+1\right)}\left(\frac{z}{2}\right)^{\nu}\ghgf 01\left(\nu+1;\frac{z^{2}}{4}\right)=e^{-\nu\pi i/2}J_{\nu}\left(e^{\pi i/2}z\right). -\] - +, +\begin_inset Formula $k>k_{0}?$ \end_inset -The problem of this approach is the insufficiently slow decay -\begin_inset Formula $\propto k^{-1}$ -\end_inset - -, so it is in fact better to compute the sum in the real space. - I have to look further. \end_layout @@ -3036,7 +2964,7 @@ where the spherical Hankel transform 2) \begin_inset Formula \[ -\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right). +\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right). \] \end_inset @@ -3046,7 +2974,7 @@ Using this convention, the inverse spherical Hankel transform is given by 3) \begin_inset Formula \[ -g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k), +g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k), \] \end_inset @@ -3059,7 +2987,7 @@ so it is not unitary. An unitary convention would look like this: \begin_inset Formula \begin{equation} -\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} +\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} \end{equation} \end_inset @@ -3113,7 +3041,7 @@ where the Hankel transform of order is defined as \begin_inset Formula \begin{equation} -\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition} +\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition} \end{equation} \end_inset