(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 46011, 872]*) (*NotebookOutlinePosition[ 46675, 895]*) (* CellTagsIndexPosition[ 46631, 891]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(f\ = \ Cos[\[Phi]]/ Sqrt[r^2 + a^2 - 2\ a\ r\ Sin[\[Theta]]\ Cos[\[Phi]]]\)], "Input"], Cell[BoxData[ \(Cos[\[Phi]]\/\@\(a\^2 + r\^2 - 2\ a\ r\ Cos[\[Phi]]\ \ Sin[\[Theta]]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]\ = \ Integrate[f, {\[Phi], 0, 2\ \[Pi]}, Assumptions\ \[Rule] \ {a^2 + r^2 + 2\ a\ r\ Sin[\[Theta]]\ > \ 0, a^2 + r^2 - 2\ a\ r\ Sin[\[Theta]]\ > \ 0\ , \ a > 0, \ r > 0, \ \[Theta]\ \[Element] \ Reals}]\)], "Input"], Cell[BoxData[ \(\(-\(\((Csc[\[Theta]]\ \((\(-\((a\^2 + r\^2)\)\)\ EllipticK[\(4\ a\ r\ \ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - \ 2\ a\ r\ Sin[\[Theta]]\) + EllipticE[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ \ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\ \((a\^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]])\) + \@\(a\^2 + r\^2 + 2\ \ a\ r\ Sin[\[Theta]]\)\ \((\(-\((a\^2 + r\^2)\)\)\ EllipticK[\(-\(\(4\ a\ r\ Sin[\ \[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)] + EllipticE[\(-\(\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \((a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]])\))\))\))\)/\((a\ r\ \@\(a\ \^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\))\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\[Alpha]]\)], "Input"], Cell[BoxData[ \(\(-\(\(2\ EllipticE[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ \ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\/\@\(a\^4 + \ r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\)\) - \(a\ Csc[\[Theta]]\ EllipticE[\ \(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]]\)]\ \ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\/\(r\ \@\(a\^4 + r\^4 + 2\ a\^2\ \ r\^2\ Cos[2\ \[Theta]]\)\) - \(r\ Csc[\[Theta]]\ EllipticE[\(4\ a\ r\ Sin[\ \[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ \ a\ r\ Sin[\[Theta]]\)\)\/\(a\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \ \[Theta]]\)\) + \(a\ Csc[\[Theta]]\ EllipticK[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\ \^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\ \[Theta]]\)\)\/\(r\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\) + \ \(r\ Csc[\[Theta]]\ EllipticK[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ \ a\ r\ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\/\(a\ \@\ \(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\) + \(2\ EllipticE[\(-\(\(4\ \ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \@\(a\ \^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]]\)\)\/\@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ \ Cos[2\ \[Theta]]\) - \(a\ Csc[\[Theta]]\ EllipticE[\(-\(\(4\ a\ r\ Sin[\ \[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \@\(a\^2 + r\^2 + \ 2\ a\ r\ Sin[\[Theta]]\)\)\/\(r\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \ \[Theta]]\)\) - \(r\ Csc[\[Theta]]\ EllipticE[\(-\(\(4\ a\ r\ Sin[\[Theta]]\)\ \/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \@\(a\^2 + r\^2 + 2\ a\ r\ \ Sin[\[Theta]]\)\)\/\(a\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\) \ + \(a\ Csc[\[Theta]]\ EllipticK[\(-\(\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + \ r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \@\(a\^2 + r\^2 + 2\ a\ r\ \ Sin[\[Theta]]\)\)\/\(r\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\) \ + \(r\ Csc[\[Theta]]\ EllipticK[\(-\(\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + \ r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \@\(a\^2 + r\^2 + 2\ a\ r\ \ Sin[\[Theta]]\)\)\/\(a\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\)\ \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(na\ = \ Numerator[\[Alpha]]\)], "Input"], Cell[BoxData[ \(\(-Csc[\[Theta]]\)\ \((\(-\((a\^2 + r\^2)\)\)\ EllipticK[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\ \^2 + 2\ a\ r\ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\) + EllipticE[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ Sin[\ \[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\ \((a\^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]])\) + \@\(a\^2 + r\^2 + 2\ a\ r\ Sin[\ \[Theta]]\)\ \((\(-\((a\^2 + r\^2)\)\)\ EllipticK[\(-\(\(4\ a\ r\ \ Sin[\[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)] + EllipticE[\(-\(\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \((a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]])\))\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(da\ = \ Denominator[\[Alpha]]\)], "Input"], Cell[BoxData[ \(a\ r\ \@\(a\^4 + r\^4 + 2\ a\^2\ r\^2\ Cos[2\ \[Theta]]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[na]\)], "Input"], Cell[BoxData[ \(\(-Csc[\[Theta]]\)\ \((\(-\((a\^2 + r\^2)\)\)\ EllipticK[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\ \^2 + 2\ a\ r\ Sin[\[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\) + EllipticE[\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 + 2\ a\ r\ Sin[\ \[Theta]]\)]\ \@\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\ \((a\^2 + r\^2 + 2\ a\ r\ Sin[\[Theta]])\) + \@\(a\^2 + r\^2 + 2\ a\ r\ Sin[\ \[Theta]]\)\ \((\(-\((a\^2 + r\^2)\)\)\ EllipticK[\(-\(\(4\ a\ r\ \ Sin[\[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)] + EllipticE[\(-\(\(4\ a\ r\ Sin[\[Theta]]\)\/\(a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]]\)\)\)]\ \((a\^2 + r\^2 - 2\ a\ r\ Sin[\[Theta]])\))\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ac\ = \ Normal[Series[\[Alpha], {r, Infinity, 6}]]\)], "Input"], Cell[BoxData[ \(\(a\ \[Pi]\ Sin[\[Theta]]\)\/r\^2 + \(\(1\/\(a\ \ r\^3\)\)\((Csc[\[Theta]]\ \((1\/2\ a\^3\ \[Pi]\ Sin[\[Theta]] + 19\/8\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 + 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) - a\ Sin[\[Theta]]\ \((\(a\^2\ \[Pi]\)\/2 + 1\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2)\) - 1\/2\ \[Pi]\ \((7\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 2\ a\ Sin[\[Theta]]\ \((9\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2 + 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\)\) + \ \(\(1\/\(a\ r\^4\)\)\((Csc[\[Theta]]\ \((\(-\(3\/8\)\)\ a\^4\ \[Pi]\ Sin[\ \[Theta]]\^2 - a\^4\ \[Pi]\ Cos[2\ \[Theta]]\ Sin[\[Theta]]\^2 + 7\/16\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 1\/2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((\(a\^2\ \[Pi]\)\/2 + 1\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2)\) + 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((7\/4\ a\^3\ Sin[\[Theta]]\^3 \ + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - a\^2\ \((9\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2 + 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(7\/4\)\)\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 2\ a\ Sin[\[Theta]]\ \((\(-\(5\/8\)\)\ a\^3\ \[Pi]\ \ Sin[\[Theta]]\^3 + 1\/2\ \[Pi]\ \((7\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + \[Pi]\ \ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - a\ Sin[\[Theta]]\ \((1\/2\ a\^3\ \[Pi]\ Sin[\[Theta]] + 1\/2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - a\ Sin[\[Theta]]\ \((a\^3\ \[Pi]\ Sin[\[Theta]] - 5\/4\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 - 1\/2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - \[Pi]\ \ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\))\)\) + \ \(\(1\/\(a\ r\^5\)\)\((Csc[\[Theta]]\ \((\(-\(9\/16\)\)\ a\^3\ \[Pi]\ Sin[\ \[Theta]]\^3\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 1\/2\ a\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((\(a\^2\ \[Pi]\)\/2 + 1\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2)\) - 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((7\/4\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ a\^2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/4\ a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(1\/2\)\)\ a\^2\ \((a\^2 \ - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ a\^2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/2\ a\^2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - a\^2\ \((\(-\(5\/8\)\)\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 + 1\/2\ \[Pi]\ \((7\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((1\/2\ a\^3\ \[Pi]\ Sin[\ \[Theta]] + 1\/2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - 1\/2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((a\^3\ \[Pi]\ Sin[\[Theta]] \ - 5\/4\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 - 1\/2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\^2\ Cos[ 2\ \[Theta]]\ \((\(-\(1\/2\)\)\ a\^3\ \[Pi]\ Sin[\[Theta]] \ - 19\/8\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 - 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(a\^2\ \[Pi]\)\/2 + 1\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2)\) + 1\/2\ \[Pi]\ \((7\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 2\ a\ Sin[\[Theta]]\ \((9\/8\ a\^2\ \[Pi]\ \ Sin[\[Theta]]\^2 + 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 3\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((305\/64\ a\^4\ \ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - 1\/2\ \[Pi]\ \((\(-\(959\/64\)\)\ a\^5\ Sin[\[Theta]]\^5 + 5\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ a\^2\ Sin[\[Theta]]\^2\ \((20\ a\^3\ Sin[\[Theta]] \ - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) - a\ Sin[\[Theta]]\ \((1\/8\ a\^4\ \[Pi]\ Sin[\[Theta]]\^2 + 1\/2\ \[Pi]\ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 \ + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) + 1\/2\ \[Pi]\ \((5831\/64\ a\^5\ Sin[\[Theta]]\^5 - 25\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 3\/4\ a\^2\ Sin[\[Theta]]\^2\ \((\(-20\)\ a\^3\ Sin[\ \[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) + 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) - 1\/2\ \[Pi]\ \((959\/64\ a\^5\ Sin[\[Theta]]\^5 - 5\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ a\^2\ Sin[\[Theta]]\^2\ \((\(-20\)\ a\^3\ Sin[\ \[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] \ + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) - 2\ a\ Sin[\[Theta]]\ \((1\/16\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2\ \ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) - 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((7\/4\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/8\ \[Pi]\ \((\(-\(1\/2\)\)\ a\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) - a\ Sin[\[Theta]]\ \((1\/2\ a\^4\ \[Pi]\ Sin[\[Theta]]\^2 - a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\ \[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/2\ \[Pi]\ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 \ + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ \[Pi]\ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) + 1\/2\ \[Pi]\ \((\(-\(5831\/64\)\)\ a\^5\ Sin[\[Theta]]\^5 + 25\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 3\/4\ a\^2\ Sin[\[Theta]]\^2\ \((20\ a\^3\ Sin[\[Theta]] \ - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) + 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] \ + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\))\)\) + \(\(1\/\(a\ r\^6\)\)\((Csc[\[Theta]]\ \ \((a\^2\ \[Pi]\ \((\(-\(a\^4\/2\)\) + 3\/2\ a\^4\ Cos[2\ \[Theta]]\^2)\)\ Sin[\[Theta]]\^2 - 1\/4\ a\ \[Pi]\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((7\/4\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/32\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2\ \((\(-\(1\/2\)\)\ a\^2\ \ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/4\ \((\(a\^2\ \[Pi]\)\/2 + 1\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2)\)\ \((\(-\(1\/2\)\)\ \ a\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/5\ a\ \[Pi]\ Sin[\[Theta]]\ \((\(-a\^3\)\ Sin[\[Theta]]\ \ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 7\/4\ a\ Sin[\[Theta]]\ \((\(-\(1\/2\)\)\ a\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\^2\ \[Pi]\ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ a\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((1\/2\ a\^3\ \[Pi]\ Sin[\ \[Theta]] + 1\/2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ a\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((a\^3\ \[Pi]\ Sin[\[Theta]] \ - 5\/4\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 - 1\/2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - 1\/2\ a\^2\ \[Pi]\ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((305\/64\ a\^4\ \ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(959\/64\)\)\ a\^5\ \ Sin[\[Theta]]\^5 + 5\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ a\^2\ Sin[\[Theta]]\^2\ \((20\ a\^3\ Sin[\[Theta]] \ - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) + 1\/2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((1\/8\ a\^4\ \[Pi]\ Sin[\ \[Theta]]\^2 + 1\/2\ \[Pi]\ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) + 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((959\/64\ a\^5\ \ Sin[\[Theta]]\^5 - 5\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ a\^2\ Sin[\[Theta]]\^2\ \((\(-20\)\ a\^3\ Sin[\ \[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] \ + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) - a\^2\ \((1\/16\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) - 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((7\/4\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/8\ \[Pi]\ \((\(-\(1\/2\)\)\ a\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) - 1\/2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((1\/2\ a\^4\ \[Pi]\ Sin[\ \[Theta]]\^2 - a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\ \[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/2\ \[Pi]\ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 \ + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ \[Pi]\ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) + a\^2\ Cos[ 2\ \[Theta]]\ \((3\/8\ a\^4\ \[Pi]\ Sin[\[Theta]]\^2 - 7\/16\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) - 1\/2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((\(a\^2\ \[Pi]\)\/2 + 1\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2)\) - 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((7\/4\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\^2\ \((9\/8\ a\^2\ \[Pi]\ Sin[\[Theta]]\^2 + 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) - a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(7\/4\)\)\ a\^3\ Sin[\ \[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 2\ a\ Sin[\[Theta]]\ \((\(-\(5\/8\)\)\ a\^3\ \[Pi]\ Sin[\ \[Theta]]\^3 + 1\/2\ \[Pi]\ \((7\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) - \ \[Pi]\ \((\(-\(1175\/64\)\)\ a\^4\ Sin[\[Theta]]\^4 + 9\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ Sin[\[Theta]]\ \((1\/2\ a\^3\ \[Pi]\ Sin[\[Theta]] + 1\/2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ Sin[\[Theta]]\ \^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ Sin[\[Theta]]\ \((a\^3\ \[Pi]\ Sin[\[Theta]] - 5\/4\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3 - 1\/2\ \[Pi]\ \((11\/4\ a\^3\ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + \ \[Pi]\ \((305\/64\ a\^4\ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) + 1\/2\ \[Pi]\ \((\(-\(105399\/256\)\)\ a\^6\ Sin[\[Theta]]\^6 + 1225\/128\ a\^4\ Sin[\[Theta]]\^4\ \((\(-8\)\ a\^2 + 80\ a\^2\ Sin[\[Theta]]\^2)\) - 25\/12\ a\^3\ Sin[\[Theta]]\^3\ \((\(-42\)\ a\^3\ Sin[\ \[Theta]] + 5\ a\ Sin[\[Theta]]\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\))\) + 9\/16\ a\^2\ Sin[\[Theta]]\^2\ \((\(-3\)\ a\^2\ \((\(-4\)\ \ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + 10\/3\ a\ Sin[\[Theta]]\ \((\(-20\)\ a\^3\ Sin[\ \[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((2\ a\^3\ Sin[\[Theta]] \ - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 2\ a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ \ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\) - 1\/2\ \[Pi]\ \((12789\/256\ a\^6\ Sin[\[Theta]]\^6 - 175\/128\ a\^4\ Sin[\[Theta]]\^4\ \((\(-8\)\ a\^2 + 80\ a\^2\ Sin[\[Theta]]\^2)\) + 5\/12\ a\^3\ Sin[\[Theta]]\^3\ \((\(-42\)\ a\^3\ Sin[\ \[Theta]] + 5\ a\ Sin[\[Theta]]\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\))\) - 3\/16\ a\^2\ Sin[\[Theta]]\^2\ \((\(-3\)\ a\^2\ \((\(-4\)\ \ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + 10\/3\ a\ Sin[\[Theta]]\ \((\(-20\)\ a\^3\ Sin[\ \[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((2\ a\^3\ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 2\ a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ \ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\) - a\ Sin[\[Theta]]\ \((\(-\(1\/2\)\)\ a\^2\ \[Pi]\ \((11\/4\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ a\^2\ \[Pi]\ \((\(-\(7\/4\)\)\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - a\ \[Pi]\ Sin[\[Theta]]\ \((305\/64\ a\^4\ \ Sin[\[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ \[Pi]\ \((\(-\(959\/64\)\)\ a\^5\ Sin[\[Theta]]\^5 \ + 5\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ a\^2\ Sin[\[Theta]]\^2\ \((20\ a\^3\ Sin[\ \[Theta]] - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ \ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\))\) \ - 1\/2\ \[Pi]\ \((5831\/64\ a\^5\ Sin[\[Theta]]\^5 - 25\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 3\/4\ a\^2\ Sin[\[Theta]]\^2\ \((\(-20\)\ a\^3\ \ Sin[\[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) + 2\ a\ Sin[\[Theta]]\ \((2\ a\^3\ \ Sin[\[Theta]] - 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\) - 2\ a\ Sin[\[Theta]]\ \((5\/16\ a\^3\ \[Pi]\ Sin[\[Theta]]\^3\ \ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ \[Pi]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\)\ \((7\/4\ a\^3\ Sin[\ \[Theta]]\^3 + a\ Sin[\[Theta]]\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 1\/8\ a\ \[Pi]\ Sin[\[Theta]]\ \((\(-\(1\/2\)\)\ a\^2\ \ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/10\ \[Pi]\ \((\(-a\^3\)\ Sin[\[Theta]]\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 7\/4\ a\ Sin[\[Theta]]\ \((\(-\(1\/2\)\)\ a\^2\ \ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\) + 5\/2\ a\^2\ Sin[\[Theta]]\^2\ \((a\^2 - a\^2\ Sin[\[Theta]]\^2)\))\))\) - 1\/2\ a\ \[Pi]\ Sin[\[Theta]]\ \((305\/64\ a\^4\ Sin[\ \[Theta]]\^4 - 3\/8\ a\^2\ Sin[\[Theta]]\^2\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) + a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + 1\/2\ \[Pi]\ \((959\/64\ a\^5\ Sin[\[Theta]]\^5 - 5\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 1\/4\ a\^2\ Sin[\[Theta]]\^2\ \((\(-20\)\ a\^3\ \ Sin[\[Theta]] + 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) - 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\ \[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\) + 1\/2\ \[Pi]\ \((\(-\(105399\/256\)\)\ a\^6\ Sin[\[Theta]]\^6 + 1225\/128\ a\^4\ Sin[\[Theta]]\^4\ \((\(-8\)\ a\^2 + 80\ a\^2\ Sin[\[Theta]]\^2)\) + 25\/12\ a\^3\ Sin[\[Theta]]\^3\ \((42\ a\^3\ \ Sin[\[Theta]] - 5\ a\ Sin[\[Theta]]\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\))\) + 9\/16\ a\^2\ Sin[\[Theta]]\^2\ \((\(-3\)\ a\^2\ \((\(-4\)\ \ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) - 10\/3\ a\ Sin[\[Theta]]\ \((20\ a\^3\ Sin[\[Theta]] \ - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((\(-2\)\ a\^3\ Sin[\[Theta]] \ + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) - 2\ a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) + 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\ \[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\) - a\ Sin[\[Theta]]\ \((1\/2\ a\^2\ \[Pi]\ \((\(-\(11\/4\)\)\ a\^3\ \ Sin[\[Theta]]\^3 + a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 1\/2\ \[Pi]\ \((\(-\(5831\/64\)\)\ a\^5\ Sin[\[Theta]]\^5 \ + 25\/8\ a\^3\ Sin[\[Theta]]\^3\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\) + 3\/4\ a\^2\ Sin[\[Theta]]\^2\ \((20\ a\^3\ Sin[\ \[Theta]] - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\) + a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) + 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\ \[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\) - 1\/2\ \[Pi]\ \((12789\/256\ a\^6\ Sin[\[Theta]]\^6 - 175\/128\ a\^4\ Sin[\[Theta]]\^4\ \((\(-8\)\ a\^2 + 80\ a\^2\ Sin[\[Theta]]\^2)\) - 5\/12\ a\^3\ Sin[\[Theta]]\^3\ \((42\ a\^3\ Sin[\[Theta]] \ - 5\ a\ Sin[\[Theta]]\ \((\(-6\)\ a\^2 + 48\ a\^2\ Sin[\[Theta]]\^2)\))\) - 3\/16\ a\^2\ Sin[\[Theta]]\^2\ \((\(-3\)\ a\^2\ \((\(-4\)\ \ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\) - 10\/3\ a\ Sin[\[Theta]]\ \((20\ a\^3\ Sin[\[Theta]] \ - 4\ a\ Sin[\[Theta]]\ \((\(-4\)\ a\^2 + 24\ a\^2\ Sin[\[Theta]]\^2)\))\))\) + a\ Sin[\[Theta]]\ \((\(-a\^2\)\ \((\(-2\)\ a\^3\ Sin[\ \[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ Sin[\[Theta]]\^2)\))\) + 2\ a\ Sin[\[Theta]]\ \((a\^2\ \((a\^2 - 4\ a\^2\ Sin[\[Theta]]\^2)\) + 2\ a\ Sin[\[Theta]]\ \((\(-2\)\ a\^3\ Sin[\ \[Theta]] + 2\ a\ Sin[\[Theta]]\ \((\(-a\^2\) + 4\ a\^2\ \ Sin[\[Theta]]\^2)\))\))\))\))\))\))\)\)\)], "Output"] }, Open ]], Cell[BoxData[""], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Ac[\([1]\)]\)], "Input"], Cell[BoxData[ \(\(a\ \[Pi]\ Sin[\[Theta]]\)\/r\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Ac[\([4]\)]]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Ac[\([3]\)]]\)], "Input"], Cell[BoxData[ \(\(-\(\(3\ a\^3\ \[Pi]\ \((3 + 5\ Cos[2\ \[Theta]])\)\ Sin[\[Theta]]\)\/\(16\ r\^4\)\)\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A\[Phi]\ = \ Ac[\([1]\)]\ + \ Simplify[Ac[\([3]\)]] + \ Simplify[Ac[\([5]\)]]\)], "Input"], Cell[BoxData[ \(\(a\ \[Pi]\ Sin[\[Theta]]\)\/r\^2 - 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\(a\^5\ \[Pi]\ P\_\(5, 1\)[\[Mu]]\)\/\(8\ \ r\^6\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({t1, t3, t5}\)], "Input"], Cell[BoxData[ \({\(-Sin[\[Theta]]\), \(-\(3\/4\)\)\ \((3\ Sin[\[Theta]] + 5\ Cos[2\ \[Theta]]\ Sin[\[Theta]])\), \(-\(15\/64\)\)\ \((15\ \ Sin[\[Theta]] + 28\ Cos[2\ \[Theta]]\ Sin[\[Theta]] + 21\ Cos[4\ \[Theta]]\ Sin[\[Theta]])\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ D[LegendreP[1, \[Mu]], \[Mu]]/LegendreP[1, 1, \[Mu]]]\)], "Input"], Cell[BoxData[ \(\(-\(1\/\@\(1 - \[Mu]\^2\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LegendreP[3, 1, \[Mu]]\)], "Input"], Cell[BoxData[ \(\(-\(3\/2\)\)\ \@\(1 - \[Mu]\^2\)\ \((\(-1\) + 5\ \[Mu]\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Integrate[ Cos[\[Phi]]\ LegendreP[3, Sin[\[Theta]] Cos[\[Phi]]], {\[Phi], 0, 2\ \[Pi]}]/Sin[\[Theta]]]\)], "Input"], Cell[BoxData[ \(\(-\(3\/16\)\)\ \[Pi]\ \((3 + 5\ Cos[2\ \[Theta]])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TrigReduce[\ LegendreP[3, 1, Cos[\[Theta]]]/Sin[\[Theta]]] /. \ Sqrt[1 - 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