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Notebook[{

Cell[CellGroupData[{
Cell["Green Function for Dirichlet Problem on Sphere", "Subtitle"],

Cell[CellGroupData[{

Cell["\<\
 \[CapitalPhi](a,\[Theta]) = V,  for 0<\[Theta]< \[Pi]/2 and  \
\[CapitalPhi](a,\[Theta]) = -V,  for \[Pi]/2<\[Theta]< \[Pi].\
\>", "Subsection"],

Cell[BoxData[
    \(Clear["\<`*\>"]\)], "Input"],

Cell[TextData[{
  "The Green function is\n\nG(r,\[Theta],\[Phi]; a,\[Psi],\[Chi]) = ",
  Cell[BoxData[
      FormBox[
        FractionBox[
          RowBox[{"a", " ", 
            RowBox[{"(", 
              RowBox[{
                FormBox[\(r\^2\),
                  "TraditionalForm"], "-", 
                FormBox[\(a\^2\),
                  "TraditionalForm"]}], 
              ")"}]}], \(\((a\^2 + r\^2 - 2\ a\ r\ x)\)\^\(3/2\)\)], 
        TraditionalForm]]],
  " \n\nwhere  x = Cos[",
  Cell[BoxData[
      \(TraditionalForm\`\[Theta]\_12\)]],
  "] = Cos[\[Theta]]Cos[\[Psi]]+Sin[\[Theta]]Sin[\[Psi]] Cos[\[Chi]-\[Phi]].\n\
 \n In our case, there is azimuthal symmetry so \[CapitalPhi] must be \
independent of the angle \[Phi]. We may, therefore, simply evaluate the \
potential at \[Phi] = 0."
}], "Text"],

Cell[BoxData[
    \(greenf\  = \ 
      a \((r^2 - a^2)\)/\((a^2 + r^2 - 2\ a\ r\ x)\)^\((3/2)\)\)], "Input"],

Cell[TextData[{
  "We can express green in terms of the ratio \[Rho]=",
  Cell[BoxData[
      \(TraditionalForm\`r\/a\)]],
  "  "
}], "Text"],

Cell[BoxData[
    \(g = \ 
      Simplify[greenf /. \ r \[Rule] \ a\ \[Rho], 
        Assumptions\  \[Rule] \ a > 0]\)], "Input"],

Cell["\<\
Expand g in an asymptotic series in \[Rho] and retain the \"n\" leading terms\
\
\>", "Text"],

Cell[BoxData[
    \(n = 10\)], "Input"],

Cell[BoxData[
    \(g = Normal[Series[g, {\[Rho], Infinity, n}]]\)], "Input"],

Cell["\<\
Substitute for the angle variable x in terms of the integration and \
observation angles\
\>", "Text"],

Cell[BoxData[
    \(x = \ 
      Cos[\[Theta]] Cos[\[Psi]]\  + \ 
        Sin[\[Theta]] Sin[\[Psi]] Cos[\[Chi]]\)], "Input"],

Cell["Expand g so that it is simpler for mathematica to integrate", "Text"],

Cell[BoxData[
    \(eg = \ Expand[g]\)], "Input"],

Cell["Do the integral over \[Chi] ", "Text"],

Cell[BoxData[
    \(\(\(\ \)\(part1 = \ Integrate[eg, {\[Chi], 0, 2  \[Pi]}]\)\)\)], "Input"],

Cell["\<\
Now, do the integral over \[Psi], divide out the factor of 4\[Pi], and expand \
the result\
\>", "Text"],

Cell[BoxData[
    \(part2\  = \ 
      Expand[\((Integrate[Sin[\[Psi]] part1, {\[Psi], 0, \[Pi]/2}] - \ 
              Integrate[
                Sin[\[Psi]] 
                  part1, {\[Psi], \[Pi]/
                    2, \[Pi]}])\)/\((4  \[Pi])\)]\)], "Input"],

Cell["\<\
Substitute to express your answer in terms of \[Mu] = Cos[\[Theta]]\
\>", "Text"],

Cell[BoxData[
    \(\[CapitalPhi]0\  = \ 
      TrigExpand[
        FullSimplify[part2 /. \ \[Theta] \[Rule] \ ArcCos[\[Mu]]]]\)], "Input"],

Cell["Pick out leading term", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(len = \ Length[\[CapitalPhi]0]\)], "Input"],

Cell[BoxData[
    \(15\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(coef\  = \ \[CapitalPhi]0[\([len]\)]/\[Mu]\)], "Input"],

Cell["Factor out the leading power", "Text"],

Cell[BoxData[
    \(res\  = \ Expand[\[CapitalPhi]0/coef]\)], "Input"],

Cell["Put the result back in the form of an asymptotic expansion", "Text"],

Cell[BoxData[
    \(res\  = \ Normal[Series[res, {\[Rho], Infinity, n}]]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(lnr\  = \ Length[res]\)], "Input"],

Cell[BoxData[
    \(5\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Express the coefficients of ",
  Cell[BoxData[
      \(TraditionalForm\`\[Rho]\^\(\(-2\) n\)\)]],
  " in terms of Legendre polynomials"
}], "Text"],

Cell[BoxData[
    \(Do[t[i] = \ 
        Simplify[\[Rho]^\((2\ i)\)\ res[\([lnr + 1 - i]\)]/
              LegendreP[2\ i + 1, \[Mu]]], {i, 1, lnr - 1}]\)], "Input"],

Cell[BoxData[
    \(\(\(Table[t[i], {i, 1, lnr - 1}]\)\(\ \)\)\)], "Input"],

Cell["Put it back together in a neat form", "Text"],

Cell[BoxData[
    \(\(\(For[{k = 1, fac = \ P\_1[\[Mu]]}, k < lnr, \(k++\), 
      fac = fac + \ 
          t[k]\ P\_\(2\ k + 1\)[\[Mu]]/\[Rho]^\((2\ k)\)]\)\(\
\[IndentingNewLine]\)
    \)\)], "Input"],

Cell[BoxData[
    \(fac\  /. \ \[Rho] \[Rule] \ r/a\)], "Input"],

Cell["\<\
Here is the resulting asymptotic expansion of \[CapitalPhi] in terms of \
Legendre Polynomials\
\>", "Text"],

Cell[BoxData[
    \(\[CapitalPhi]\  = 
      V\ \ Expand\ [coef\ fac\  /. \ \[Rho] \[Rule] \ r/a]\)], "Input"],

Cell[BoxData[
    \(V \((Normal[
          Series[\((1 - \((z^2 - a^2)\)/\((z\ Sqrt[z^2 + a^2])\))\), {z, 
              Infinity, 10}]])\)\)], "Input"]
}, Open  ]]
}, Open  ]]
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