(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 31544, 960] NotebookOptionsPosition[ 29640, 894] NotebookOutlinePosition[ 29977, 909] CellTagsIndexPosition[ 29934, 906] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\"\\"", "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{{3.4330043905364647`*^9, 3.433004508967387*^9}, { 3.433004832067649*^9, 3.433004848956999*^9}, {3.4330054256075974`*^9, 3.43300543142948*^9}, {3.43300564693311*^9, 3.433005652560863*^9}, { 3.433095422785439*^9, 3.433095435915572*^9}, {3.433098549681946*^9, 3.433098554535985*^9}}], Cell[BoxData["\<\"Problem 3.1\"\>"], "Output", CellChangeTimes->{3.4331170222190866`*^9, 3.433117569694275*^9, 3.433183989612178*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"a", ",", "b", ",", "V", ",", "Q"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\"\< for even L \>\"", ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"eq1", "=", RowBox[{ RowBox[{"V", " ", RowBox[{"Q", "[", "L", "]"}]}], "==", RowBox[{ RowBox[{"2", "/", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "L"}], "+", "1"}], ")"}]}], RowBox[{"(", RowBox[{ RowBox[{"A", " ", SuperscriptBox["a", "L"]}], "+", RowBox[{"B", " ", SuperscriptBox["a", RowBox[{ RowBox[{"-", "L"}], "-", "1"}]]}]}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"eq2", "=", RowBox[{ RowBox[{"V", " ", RowBox[{"Q", "[", "L", "]"}]}], "==", RowBox[{ RowBox[{"2", "/", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "L"}], "+", "1"}], ")"}]}], RowBox[{"(", RowBox[{ RowBox[{"A", " ", SuperscriptBox["b", "L"]}], "+", RowBox[{"B", " ", SuperscriptBox["b", RowBox[{ RowBox[{"-", "L"}], "-", "1"}]]}]}], ")"}]}]}]}]}], "Input", CellChangeTimes->{{3.4330043905364647`*^9, 3.433004508967387*^9}, { 3.433004832067649*^9, 3.433004848956999*^9}, {3.4330054256075974`*^9, 3.43300543142948*^9}, {3.43300564693311*^9, 3.433005652560863*^9}, { 3.433095422785439*^9, 3.433095435915572*^9}, {3.433098549681946*^9, 3.433098577087569*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"V", " ", RowBox[{"Q", "[", "L", "]"}]}], "\[Equal]", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["a", "L"], " ", "A"}], "+", RowBox[{ SuperscriptBox["a", RowBox[{ RowBox[{"-", "1"}], "-", "L"}]], " ", "B"}]}], ")"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}]], "Output", CellChangeTimes->{ 3.433004544999176*^9, {3.433004845805373*^9, 3.433004849443817*^9}, 3.43300509768686*^9, 3.433005148221498*^9, 3.433005653436881*^9, 3.43301201237379*^9, 3.433095440800256*^9, 3.433097411140291*^9, { 3.433098566252344*^9, 3.4330985782431507`*^9}, 3.433117022371953*^9, 3.433117569843952*^9, 3.4331839902691936`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"V", " ", RowBox[{"Q", "[", "L", "]"}]}], "\[Equal]", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"A", " ", SuperscriptBox["b", "L"]}], "+", RowBox[{ SuperscriptBox["b", RowBox[{ RowBox[{"-", "1"}], "-", "L"}]], " ", "B"}]}], ")"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}]], "Output", CellChangeTimes->{ 3.433004544999176*^9, {3.433004845805373*^9, 3.433004849443817*^9}, 3.43300509768686*^9, 3.433005148221498*^9, 3.433005653436881*^9, 3.43301201237379*^9, 3.433095440800256*^9, 3.433097411140291*^9, { 3.433098566252344*^9, 3.4330985782431507`*^9}, 3.433117022371953*^9, 3.433117569843952*^9, 3.433183990273364*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"FullSimplify", "[", RowBox[{"Flatten", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{"eq1", ",", "eq2"}], "}"}], ",", RowBox[{"{", RowBox[{"A", ",", "B"}], "}"}]}], "]"}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Aeven", "[", "L_", "]"}], "=", RowBox[{"A", " ", "/.", " ", "%"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Beven", "[", "L_", "]"}], "=", RowBox[{"B", " ", "/.", " ", "%%"}]}]}], "Input", CellChangeTimes->{{3.433004547589567*^9, 3.433004587343018*^9}, { 3.433005081174699*^9, 3.4330050917302*^9}, {3.433005143996426*^9, 3.433005265503542*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"A", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], "-", SuperscriptBox["b", RowBox[{"1", "+", "L"}]]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]}], ",", RowBox[{"B", "\[Rule]", FractionBox[ RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], " ", SuperscriptBox["b", RowBox[{"1", "+", "L"}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "L"], "-", SuperscriptBox["b", "L"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]}]}], "}"}]], "Output", CellChangeTimes->{{3.4330045658192167`*^9, 3.4330045890121326`*^9}, { 3.433005092846394*^9, 3.43300510408284*^9}, 3.433005152387179*^9, { 3.433005183051346*^9, 3.433005235445648*^9}, 3.43300526620851*^9, 3.433005657793153*^9, 3.433012013456233*^9, 3.433097411780726*^9, 3.4330985828689003`*^9, 3.433117024060779*^9, 3.4331175702012444`*^9, 3.433183990723589*^9}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], "-", SuperscriptBox["b", RowBox[{"1", "+", "L"}]]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]], "Output", CellChangeTimes->{{3.4330045658192167`*^9, 3.4330045890121326`*^9}, { 3.433005092846394*^9, 3.43300510408284*^9}, 3.433005152387179*^9, { 3.433005183051346*^9, 3.433005235445648*^9}, 3.43300526620851*^9, 3.433005657793153*^9, 3.433012013456233*^9, 3.433097411780726*^9, 3.4330985828689003`*^9, 3.433117024060779*^9, 3.4331175702012444`*^9, 3.433183990730877*^9}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], " ", SuperscriptBox["b", RowBox[{"1", "+", "L"}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "L"], "-", SuperscriptBox["b", "L"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]], "Output", CellChangeTimes->{{3.4330045658192167`*^9, 3.4330045890121326`*^9}, { 3.433005092846394*^9, 3.43300510408284*^9}, 3.433005152387179*^9, { 3.433005183051346*^9, 3.433005235445648*^9}, 3.43300526620851*^9, 3.433005657793153*^9, 3.433012013456233*^9, 3.433097411780726*^9, 3.4330985828689003`*^9, 3.433117024060779*^9, 3.4331175702012444`*^9, 3.433183990733883*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\"\< for odd L \>\"", ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"eq1", "=", RowBox[{ RowBox[{"V", " ", RowBox[{"Q", "[", "L", "]"}]}], "==", RowBox[{ RowBox[{"2", "/", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "L"}], "+", "1"}], ")"}]}], RowBox[{"(", RowBox[{ RowBox[{"A", " ", SuperscriptBox["a", "L"]}], "+", RowBox[{"B", " ", SuperscriptBox["a", RowBox[{ RowBox[{"-", "L"}], "-", "1"}]]}]}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"eq2", "=", RowBox[{ RowBox[{ RowBox[{"-", "V"}], " ", RowBox[{"Q", "[", "L", "]"}]}], "==", RowBox[{ RowBox[{"2", "/", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "L"}], "+", "1"}], ")"}]}], RowBox[{"(", RowBox[{ RowBox[{"A", " ", SuperscriptBox["b", "L"]}], "+", RowBox[{"B", " ", SuperscriptBox["b", RowBox[{ RowBox[{"-", "L"}], "-", "1"}]]}]}], ")"}]}]}]}]}], "Input", CellChangeTimes->{ 3.4330049068921623`*^9, {3.433005116129343*^9, 3.433005118188718*^9}, { 3.4330056668459177`*^9, 3.433005671569512*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"V", " ", RowBox[{"Q", "[", "L", "]"}]}], "\[Equal]", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["a", "L"], " ", "A"}], "+", RowBox[{ SuperscriptBox["a", RowBox[{ RowBox[{"-", "1"}], "-", "L"}]], " ", "B"}]}], ")"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}]], "Output", CellChangeTimes->{3.433004908067111*^9, 3.433005123475107*^9, 3.43300527919665*^9, 3.433005672556576*^9, 3.433012013584193*^9, 3.433097411842905*^9, 3.433098586288275*^9, 3.433117024178164*^9, 3.433117570259276*^9, 3.433183990841481*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "V"}], " ", RowBox[{"Q", "[", "L", "]"}]}], "\[Equal]", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"A", " ", SuperscriptBox["b", "L"]}], "+", RowBox[{ SuperscriptBox["b", RowBox[{ RowBox[{"-", "1"}], "-", "L"}]], " ", "B"}]}], ")"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}]], "Output", CellChangeTimes->{3.433004908067111*^9, 3.433005123475107*^9, 3.43300527919665*^9, 3.433005672556576*^9, 3.433012013584193*^9, 3.433097411842905*^9, 3.433098586288275*^9, 3.433117024178164*^9, 3.433117570259276*^9, 3.433183990852233*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"FullSimplify", "[", RowBox[{"Flatten", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{"eq1", ",", "eq2"}], "}"}], ",", RowBox[{"{", RowBox[{"A", ",", "B"}], "}"}]}], "]"}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Aodd", "[", "L_", "]"}], "=", RowBox[{"A", " ", "/.", " ", "%"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Bodd", "[", "L_", "]"}], "=", RowBox[{"B", " ", "/.", " ", "%%"}]}]}], "Input", CellChangeTimes->{{3.433005130303294*^9, 3.433005131931065*^9}, { 3.433005282810176*^9, 3.433005323808579*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"A", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], "+", SuperscriptBox["b", RowBox[{"1", "+", "L"}]]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]}], ",", RowBox[{"B", "\[Rule]", RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], " ", SuperscriptBox["b", RowBox[{"1", "+", "L"}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "L"], "+", SuperscriptBox["b", "L"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.4330049726816387`*^9, {3.4330052947508383`*^9, 3.4330053246291943`*^9}, 3.433005681256024*^9, 3.43301201389445*^9, 3.433097412149001*^9, 3.433098591179839*^9, 3.4331170244178267`*^9, 3.433117570499189*^9, 3.433183991091977*^9}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], "+", SuperscriptBox["b", RowBox[{"1", "+", "L"}]]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]], "Output", CellChangeTimes->{ 3.4330049726816387`*^9, {3.4330052947508383`*^9, 3.4330053246291943`*^9}, 3.433005681256024*^9, 3.43301201389445*^9, 3.433097412149001*^9, 3.433098591179839*^9, 3.4331170244178267`*^9, 3.433117570499189*^9, 3.433183991095264*^9}], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", "L"}]], " ", SuperscriptBox["b", RowBox[{"1", "+", "L"}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "L"], "+", SuperscriptBox["b", "L"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}], ")"}], " ", "V", " ", RowBox[{"Q", "[", "L", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]], "-", SuperscriptBox["b", RowBox[{"1", "+", RowBox[{"2", " ", "L"}]}]]}], ")"}]}]]}]], "Output", CellChangeTimes->{ 3.4330049726816387`*^9, {3.4330052947508383`*^9, 3.4330053246291943`*^9}, 3.433005681256024*^9, 3.43301201389445*^9, 3.433097412149001*^9, 3.433098591179839*^9, 3.4331170244178267`*^9, 3.433117570499189*^9, 3.433183991141726*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"LegendreP", "[", RowBox[{"n", ",", "x"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{ 3.433005709365288*^9, {3.4330064566886377`*^9, 3.433006468945129*^9}, { 3.433006554535623*^9, 3.433006669320733*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", FractionBox["1", "2"], ",", "0", ",", RowBox[{"-", FractionBox["1", "8"]}], ",", "0", ",", FractionBox["1", "16"], ",", "0", ",", RowBox[{"-", FractionBox["5", "128"]}], ",", "0", ",", FractionBox["7", "256"], ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.4330066703208313`*^9, 3.433012015043476*^9, 3.433097413086019*^9, 3.4330986036472063`*^9, 3.433117025945553*^9, 3.433117570878871*^9, 3.4331839914980297`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\"\< from Eq. (3.26) in text; only for odd L\>\"", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Q", "[", "L_", "]"}], "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "/", "2"}], ")"}], RowBox[{ RowBox[{"(", RowBox[{"L", "-", "1"}], ")"}], "/", "2"}]], " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"L", "-", "2"}], ")"}], "!!"}], "/", "2"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"L", "+", "1"}], ")"}], "/", "2"}], ")"}], "!"}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"\"\< for L=0 \>\"", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Q", "[", "0", "]"}], "=", "1"}], ";"}]}], "Input", CellChangeTimes->{ 3.433005709365288*^9, {3.4330064566886377`*^9, 3.433006468945129*^9}, { 3.433006554535623*^9, 3.433006683187278*^9}, {3.433006907257576*^9, 3.433006910733292*^9}, {3.433006968510305*^9, 3.43300697266157*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["\[ImaginaryI]", RowBox[{ RowBox[{"-", "1"}], "+", "L"}]], " ", SuperscriptBox["2", RowBox[{ RowBox[{"-", "1"}], "+", FractionBox[ RowBox[{"1", "-", "L"}], "2"]}]], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "L"}], ")"}], "!!"}]}], RowBox[{ FractionBox[ RowBox[{"1", "+", "L"}], "2"], "!"}]]], "Output", CellChangeTimes->{ 3.433006648417124*^9, 3.4330066841652327`*^9, {3.4330069079599323`*^9, 3.433006911411026*^9}, 3.433006973459717*^9, 3.433012015171887*^9, 3.433097413144812*^9, 3.433098610468093*^9, 3.43311702607743*^9, 3.433117571023903*^9, 3.4331839915566273`*^9}] }, Open ]], Cell[BoxData[ RowBox[{"\"\< so for L=0 \>\"", ";"}]], "Input", CellChangeTimes->{{3.433005717599958*^9, 3.433005752606304*^9}, { 3.4330058396132603`*^9, 3.433005930283389*^9}, {3.433005980678533*^9, 3.4330059808538*^9}, {3.433006486060504*^9, 3.4330065282085543`*^9}, { 3.433006609831275*^9, 3.433006610354189*^9}, {3.433006690302125*^9, 3.43300669994715*^9}, {3.433006936019685*^9, 3.433006946885132*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Aeven", "[", "0", "]"}], "\[IndentingNewLine]", RowBox[{"Beven", "[", "0", "]"}]}], "Input", CellChangeTimes->{{3.433006993806218*^9, 3.433007015808372*^9}}], Cell[BoxData[ FractionBox["V", "2"]], "Output", CellChangeTimes->{{3.433006998614502*^9, 3.433007016572836*^9}, 3.4330120152651377`*^9, 3.43309741324121*^9, 3.433098613688855*^9, 3.433098868924354*^9, 3.4331170261762247`*^9, 3.433117571111113*^9, 3.433183991640358*^9}], Cell[BoxData["0"], "Output", CellChangeTimes->{{3.433006998614502*^9, 3.433007016572836*^9}, 3.4330120152651377`*^9, 3.43309741324121*^9, 3.433098613688855*^9, 3.433098868924354*^9, 3.4331170261762247`*^9, 3.433117571111113*^9, 3.433183991665292*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"FullSimplify", "[", RowBox[{"Aodd", "[", "k", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "3", ",", "2"}], "}"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Table", "[", RowBox[{ RowBox[{"FullSimplify", "[", RowBox[{"Bodd", "[", "k", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "3", ",", "2"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.43300705701337*^9, 3.4330071026936483`*^9}, { 3.433007404119652*^9, 3.433007407194579*^9}, {3.433012033625044*^9, 3.433012163513131*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"3", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]}], ")"}], " ", "V"}], RowBox[{"4", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "3"], "-", SuperscriptBox["b", "3"]}], ")"}]}]], ",", RowBox[{"-", FractionBox[ RowBox[{"7", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "4"]}], ")"}], " ", "V"}], RowBox[{"16", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "7"], "-", SuperscriptBox["b", "7"]}], ")"}]}]]}]}], "}"}]], "Output", CellChangeTimes->{{3.433007060774207*^9, 3.433007103454212*^9}, 3.4330074083365717`*^9, 3.433012015443879*^9, 3.433012164628057*^9, { 3.433097382381467*^9, 3.433097413578308*^9}, 3.433098524004047*^9, 3.4330986166173773`*^9, 3.4331170265248623`*^9, 3.433117571312298*^9, 3.433183991842812*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], " ", "V"}], RowBox[{"4", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "3"], "-", SuperscriptBox["b", "3"]}], ")"}]}]]}], ",", FractionBox[ RowBox[{"7", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["b", "4"], " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "3"], "+", SuperscriptBox["b", "3"]}], ")"}], " ", "V"}], RowBox[{"16", " ", RowBox[{"(", RowBox[{ SuperscriptBox["a", "7"], "-", SuperscriptBox["b", "7"]}], ")"}]}]]}], "}"}]], "Output", CellChangeTimes->{{3.433007060774207*^9, 3.433007103454212*^9}, 3.4330074083365717`*^9, 3.433012015443879*^9, 3.433012164628057*^9, { 3.433097382381467*^9, 3.433097413578308*^9}, 3.433098524004047*^9, 3.4330986166173773`*^9, 3.4331170265248623`*^9, 3.433117571312298*^9, 3.433183991898234*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\"\\"", ";"}], "\[IndentingNewLine]", RowBox[{"r", "=", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}], "]"}], " ", "/.", " ", RowBox[{"y", "\[Rule]", "0"}]}]}], "\[IndentingNewLine]", RowBox[{"Cos\[Theta]", "=", RowBox[{ RowBox[{"z", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}], "]"}]}], " ", "/.", " ", RowBox[{"y", "\[Rule]", "0"}]}]}]}], "Input", CellChangeTimes->{{3.433095503359909*^9, 3.43309554390549*^9}, { 3.433096922743548*^9, 3.433096934238071*^9}, {3.433097079769999*^9, 3.433097107284554*^9}, {3.433097171062943*^9, 3.433097201794284*^9}, { 3.433097241216724*^9, 3.433097296335471*^9}}], Cell[BoxData[ SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]]], "Output", CellChangeTimes->{ 3.433097093414178*^9, 3.433097196427162*^9, {3.433097291243678*^9, 3.433097297226719*^9}, 3.433097413712644*^9, 3.4330980865070677`*^9, 3.4330986237707787`*^9, 3.433117026647121*^9, 3.433117571439288*^9, 3.433183992011548*^9}], Cell[BoxData[ FractionBox["z", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["z", "2"]}]]]], "Output", CellChangeTimes->{ 3.433097093414178*^9, 3.433097196427162*^9, {3.433097291243678*^9, 3.433097297226719*^9}, 3.433097413712644*^9, 3.4330980865070677`*^9, 3.4330986237707787`*^9, 3.433117026647121*^9, 3.433117571439288*^9, 3.433183992039207*^9}] }, Open ]], Cell[BoxData[{ RowBox[{ RowBox[{"V", "=", "1"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a", "=", "1"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"b", "=", "2"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Lmax", "=", "21"}], ";"}]}], "Input", CellChangeTimes->{{3.433095503359909*^9, 3.43309554390549*^9}, { 3.433096922743548*^9, 3.433096934238071*^9}, {3.433097305314785*^9, 3.433097338379603*^9}, {3.433097428914031*^9, 3.433097708720153*^9}, { 3.4330979380114717`*^9, 3.4330979392941236`*^9}, 3.433098371051114*^9, { 3.4330986659047117`*^9, 3.433098667134927*^9}, {3.4330990842172728`*^9, 3.433099086908119*^9}, 3.4331056264080553`*^9, {3.43311713062538*^9, 3.433117132207218*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"phi", "[", RowBox[{"x_", ",", "z_"}], "]"}], "=", RowBox[{"N", "[", RowBox[{ RowBox[{ RowBox[{"Aeven", "[", "0", "]"}], " ", RowBox[{"LegendreP", "[", RowBox[{"0", ",", "Cos\[Theta]"}], "]"}]}], "+", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Aodd", "[", "L", "]"}], " ", SuperscriptBox["r", "L"]}], " ", "+", RowBox[{ RowBox[{"Bodd", "[", "L", "]"}], " ", RowBox[{"1", "/", SuperscriptBox["r", RowBox[{"L", "+", "1", " "}]]}]}]}], ")"}], " ", RowBox[{"LegendreP", "[", RowBox[{"L", ",", "Cos\[Theta]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"L", ",", "1", ",", "Lmax", ",", "2"}], "}"}]}], "]"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.433095503359909*^9, 3.43309554390549*^9}, { 3.433096922743548*^9, 3.433096934238071*^9}, {3.433097305314785*^9, 3.433097338379603*^9}, {3.433097428914031*^9, 3.433097716653048*^9}, { 3.433098099346674*^9, 3.433098099630969*^9}, {3.433098387410529*^9, 3.433098391306807*^9}, 3.433098818521649*^9, {3.433098883097426*^9, 3.433098904228442*^9}, {3.433098938249929*^9, 3.433098954405857*^9}, 3.4330990907371883`*^9}], Cell[BoxData[ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{"phi", "[", RowBox[{"x", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"RegionFunction", "\[Rule]", RowBox[{"Function", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "z"}], "}"}], ",", RowBox[{"a", "<=", SqrtBox[ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}]], "\[LessEqual]", "b"}]}], " ", "]"}]}], ",", RowBox[{"Contours", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "0.1", ",", "0.25", ",", "0.5", ",", "1"}], "}"}]}], ",", RowBox[{"ContourStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"Green", ",", RowBox[{"Thickness", "[", "0.01", "]"}]}], "}"}], ",", "Purple", ",", "Cyan", ",", "Yellow", ",", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"Thickness", "[", "0.01", "]"}]}], "}"}]}], "}"}]}], ",", RowBox[{"ContourShading", "\[Rule]", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.433097744886601*^9, 3.433097759077581*^9}, { 3.433097803913102*^9, 3.4330978164230127`*^9}, {3.433097862264883*^9, 3.4330978977825193`*^9}, {3.433099321913484*^9, 3.433099323426607*^9}, { 3.433099418320986*^9, 3.4330994878774137`*^9}, {3.433099630461341*^9, 3.433099659765263*^9}, {3.43311707197326*^9, 3.433117104530973*^9}, { 3.433117135569656*^9, 3.433117140400725*^9}, {3.4331171877578077`*^9, 3.4331171917409697`*^9}, {3.433117235108324*^9, 3.433117242809112*^9}, { 3.433117289545598*^9, 3.433117334065415*^9}, {3.43311736517241*^9, 3.433117422328384*^9}, {3.433117961674561*^9, 3.433117973637422*^9}, { 3.433118034570857*^9, 3.433118049823741*^9}, {3.433118098300249*^9, 3.4331181093840446`*^9}, {3.433118143904191*^9, 3.433118145293593*^9}, { 3.4331182104925632`*^9, 3.433118212625621*^9}, {3.433118247540614*^9, 3.433118266209546*^9}, {3.433118327108695*^9, 3.4331183459698668`*^9}, { 3.4331183898400173`*^9, 3.43311841036385*^9}, {3.4331184603205843`*^9, 3.433118471261756*^9}, {3.433118578769095*^9, 3.4331186077972603`*^9}, { 3.433118653829101*^9, 3.433118653978486*^9}, {3.433118689533287*^9, 3.433118708970128*^9}, {3.43311875108689*^9, 3.433118780110983*^9}, { 3.433118812780109*^9, 3.433118835716405*^9}, {3.4331188691642714`*^9, 3.433118912340723*^9}}], Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"phi", "[", RowBox[{"x", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"RegionFunction", "\[Rule]", RowBox[{"Function", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "z"}], "}"}], ",", RowBox[{"a", "<", SqrtBox[ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}]], "<", "b"}]}], "]"}]}]}], "]"}]], "Input",\ CellChangeTimes->{{3.433105694145241*^9, 3.433105702431966*^9}, { 3.433117527566465*^9, 3.4331175462592087`*^9}, {3.433117764923333*^9, 3.4331178232712393`*^9}, {3.433118843087467*^9, 3.433118846470273*^9}}] }, WindowSize->{999, 974}, WindowMargins->{{149, Automatic}, {Automatic, 36}}, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (June 19, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 390, 6, 43, "Input"], Cell[983, 31, 137, 2, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[1157, 38, 1466, 45, 98, "Input"], Cell[2626, 85, 773, 21, 49, "Output"], Cell[3402, 108, 771, 21, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4210, 134, 669, 18, 58, "Input"], Cell[4882, 154, 1833, 54, 56, "Output"], Cell[6718, 210, 970, 27, 56, "Output"], Cell[7691, 239, 1036, 29, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8764, 273, 1218, 42, 83, "Input"], Cell[9985, 317, 668, 19, 49, "Output"], Cell[10656, 338, 687, 20, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11380, 363, 620, 17, 58, "Input"], Cell[12003, 382, 1758, 54, 56, "Output"], Cell[13764, 438, 852, 26, 56, "Output"], Cell[14619, 466, 958, 29, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15614, 500, 475, 13, 27, "Input"], Cell[16092, 515, 504, 12, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16633, 532, 1062, 32, 77, "Input"], Cell[17698, 566, 728, 22, 64, "Output"] }, Open ]], Cell[18441, 591, 418, 6, 27, "Input"], Cell[CellGroupData[{ Cell[18884, 601, 185, 3, 43, "Input"], Cell[19072, 606, 283, 5, 43, "Output"], Cell[19358, 613, 263, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19658, 622, 617, 16, 43, "Input"], Cell[20278, 640, 1022, 30, 56, "Output"], Cell[21303, 672, 1112, 32, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22452, 709, 891, 22, 58, "Input"], Cell[23346, 733, 373, 9, 38, "Output"], Cell[23722, 744, 397, 10, 54, "Output"] }, Open ]], Cell[24134, 757, 741, 15, 73, "Input"], Cell[24878, 774, 1375, 35, 33, "Input"], Cell[26256, 811, 2539, 55, 80, "Input"], Cell[28798, 868, 838, 24, 39, "Input"] } ] *) (* End of internal cache information *)