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http://functions.wolfram.com/07.37.06.0016.01
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SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]]
SphericalHarmonicY[n, m, \[CurlyTheta], -\[CurlyPhi]] ==
((2 n + 1)/(4 Pi)) Sum[(-1)^(k + m) ((k + n)!/(-k + n)!)
((2 k - 1)!!/(2 k)!!) (Sin[\[CurlyTheta]]^(2 k)/((k - m)! (k + m)!)),
{k, Abs[m], n}] /; Element[n, Integers] && n >= 0 &&
Element[m, Integers] && Abs[m] <= n
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", RowBox[List["-", "\[CurlyPhi]"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Abs", "[", "m", "]"]]]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "m"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "n"]], ")"]], "!"]]], " ", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]], "!!"]], RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!!"]]], " ", FractionBox[SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], RowBox[List["2", "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mrow> <mo> - </mo> <mi> φ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> m </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mi> sin </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> m </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> φ </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <abs /> <ci> m </ci> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Factorial2 </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <sin /> <ci> ϑ </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <leq /> <apply> <abs /> <ci> m </ci> </apply> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", RowBox[List["-", "\[CurlyPhi]_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Abs", "[", "m", "]"]]]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "m"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!!"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m"]], ")"]], "!"]]]], ")"]]]]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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