[Date Index]
[Thread Index]
[Author Index]
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
____________________________________________________________________
Prev by Date:
**Re: 2d-interpolation on non-equidistant data grid ?**
Next by Date:
**Question regarding expressions with Sum[...]**
Previous by thread:
**Apparent Documentation Error for SphericalHarmonicY**
Next by thread:
**Re: Apparent Documentation Error for SphericalHarmonicY**
| |