Re: Apparent Documentation Error for SphericalHarmonicY
- To: mathgroup at smc.vnet.net
- Subject: [mg7801] Re: Apparent Documentation Error for SphericalHarmonicY
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Tue, 8 Jul 1997 22:41:13 -0400
- Organization: University of Western Australia
- Sender: owner-wri-mathgroup at wolfram.com
BobHanlon at aol.com wrote: > SphericalHarmonicY[n, m, theta, phi] // FunctionExpand > > includes a factor of > > Sqrt[Gamma[n-m+1]/Gamma[n+m+1]] > > whereas the on-line documentation and the Mathematica Reference Guide reflect > the reciprocal factor, i.e., > > Sqrt[(n+m)!/(n-m)!] > > Assuming that the implementation was checked more rigorously than the > documentation, the documentation needs to be corrected. You are correct and the documentation is incorrect as the following Notebook shows: Notebook[{ Cell["On page 1202 the Reference Guide documentation states that", "Text"], Cell[TextData[{ " \[FilledSmallSquare] For ", Cell[BoxData[ \(TraditionalForm\`l \[GreaterEqual] 0\)], "InlineFormula"], ", ", Cell[BoxData[ \(TraditionalForm \`\(Y\_l\%m\)(\[Theta], \[Phi]) = \(\ at \(\((2 l + 1)\)/\((4 \[Pi])\)\)\) \(\ at \(\(\((l + m)\)!\)/\(\((l - m)\)!\)\)\) \(\(P\_l\%m\)(cos(\[Theta]))\) e\^\(i m \[Phi]\)\)], "InlineFormula"], " where ", Cell[BoxData[ \(TraditionalForm\`P\_l\%m\)], "InlineFormula"], " is the associated Legendre function. " }], "Text"], Cell["This should read", "Text"], Cell[TextData[{ " \[FilledSmallSquare] For ", Cell[BoxData[ \(TraditionalForm\`l \[GreaterEqual] 0\)], "InlineFormula"], ", ", Cell[BoxData[ FormBox[ RowBox[{\(\(Y\_l\%m\)(\[Theta], \[Phi])\), "=", RowBox[{ \(\ at \(\((2 l + 1)\)/\((4 \[Pi])\)\)\), \(\ at \(\(\((l - m)\)!\)/\(\((l + m)\)!\)\)\), RowBox[{ SubsuperscriptBox[ TagBox["P", LegendreP], "l", "m"], "(", \(cos(\[Theta])\), ")"}], \(\[ExponentialE]\^\(\[ImaginaryI]\ m\ \[Phi]\)\)}]}], TraditionalForm]], "InlineFormula"], " where ", Cell[BoxData[ \(TraditionalForm\`P\_l\%m\)], "InlineFormula"], " is the associated Legendre function. " }], "Text"], Cell["\<\ (which agrees with Angular Momentum in Quantum Mechanics by Edmonds \ apart from a phase factor) as can be seen from\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{\(SphericalHarmonicY[l, m, \[Theta], \[Phi]]\), "-", RowBox[{ \(\ at \(\((2 l + 1)\)/\((4 \[Pi])\)\)\), \(\ at \(\(\((l - m)\)!\)/\(\((l + m)\)!\)\)\), RowBox[{ SubsuperscriptBox[ TagBox["P", LegendreP], "l", "m"], "(", \(cos(\[Theta])\), ")"}], \(\[ExponentialE]\^\(\[ImaginaryI]\ m\ \[Phi]\)\)}]}], ",", \({l, 0, 2}\), ",", \({m, \(-l\), l}\)}], "]"}], "//", "Simplify"}], "//", "PowerExpand"}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`{{0}, {0, 0, 0}, {0, 0, 0, 0, 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "We can use ", StyleBox["PowerExpand", "Input"], " because ", Cell[BoxData[ \(TraditionalForm \`\ at \(\(sin\^2\)(\[Theta])\) \[Equal] sin(\[Theta])\)]], " for ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\ \[Epsilon]\ [0, \[Pi]]\)]], "." }], "Text"] } ] Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul at physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/Paul God IS a weakly left-handed dice player ____________________________________________________________________