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MathGroup Archive 1997

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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
____________________________________________________________________


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