(************** 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). 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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[ 19970, 511]*) (*NotebookOutlinePosition[ 20626, 534]*) (* CellTagsIndexPosition[ 20582, 530]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "\[Zeta] and \[Gamma] represent choices for the spin gauge. \[Gamma] is \ defined by the choice of complex vector ", Cell[BoxData[ \(TraditionalForm\`m\)]], ", where \n ", Cell[BoxData[ \(TraditionalForm\`m \[Congruent] \(\[ExponentialE]\^\[ImaginaryI]\ \[Gamma]\/\@2\) \((\(\[Theta]\&^\) + \[ImaginaryI]\ \(\[Phi]\&^\))\)\)]], ",\nwith ", Cell[BoxData[ \(TraditionalForm\`\(\[Theta]\&^\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[Phi]\&^\)\)]], " being the usual unit basis vectors in spherical coordinates. Ideally, \ the choice made in this definition would also fix the choice of \[Zeta], as \ \[Zeta]=\[Gamma]. Unfortunately, this is not always the chosen convention, \ and \[Zeta] is simply chosen as the parameter which makes the definition \ below correct.\n\nBy default, these parameters are kept unspecified by the \ entry below. The \"classic\" (though not universal) convention is to choose \ \[Zeta]=\[Gamma]=0. [Note that \[Gamma] is not used in this notebook.]" }], "Text"], Cell[BoxData[{ \(\(\[Zeta] =. ;\)\), "\[IndentingNewLine]", \(\(\[Gamma] =. ;\)\)}], "Input"], Cell[CellGroupData[{ Cell["Do This First!!!", "Section"], Cell["These are just a few useful commands to Mathematica.", "Text"], Cell[BoxData[{ \(Off[General::"\"]\), "\[IndentingNewLine]", \(\($PreRead = \((# /. {"\" \[Rule] "\", "\" \ \[Rule] "\"})\) &;\)\), "\[IndentingNewLine]", \(MakeBoxes[MyVariableI, form : \((StandardForm | TraditionalForm)\)] := "\"\), "\[IndentingNewLine]", \(MakeBoxes[MyVariableC, form : \((StandardForm | TraditionalForm)\)] := "\"\), "\[IndentingNewLine]", \(<< Graphics`MultipleListPlot`\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Spin-Weighted Spherical Harmonics Calculator", "Section"], Cell[CellGroupData[{ Cell["Function Definition", "Subsection"], Cell[BoxData[ \(\(Y[s_, l_, m_, \[Theta]_, \[Phi]_] := \(\[ExponentialE]\^\(\[ImaginaryI]\ s\ \ \[Zeta]\)\) \(\((\(-1\))\)\^m\) Simplify[\(\@\(\(\(\((l + m)\)!\) \(\((l - m)\)!\) \((2 l + 1)\)\)\/\(\(\((l + s)\)!\) \(\((l - s)\)!\) 4 \[Pi]\)\)\) \(\((Sin[\[Theta]\/2])\)\^\(2 l\)\) \(\[Sum]\_\(r = 0\)\%\(l - s\)\((Binomial[l - s, r] Binomial[l + s, r + s - m] \(\((\(-1\))\)\^\(l - r - s\)\) \(\[ExponentialE]\^\(\[ImaginaryI]\ m\ \[Phi]\ \)\) \((Cot[\[Theta]\/2])\)\^\(2 r + s - m\))\)\), Assumptions \[Rule] {\[Phi] \[Element] Reals, \[Theta] \[Element] Reals}];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Tests (s=0,1,2; 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