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http://functions.wolfram.com/07.08.23.0002.01
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Sum[((n - m)!/(n + m)!) LegendreP[n, m, 2, Cos[\[Theta]]]
LegendreP[n, m, 2, Cos[Subscript[\[Theta], 1]]]
Cos[m (\[Phi] - Subscript[\[Phi], 1])], {m, -n, n}] ==
LegendreP[n, Cos[\[Theta]] Cos[Subscript[\[Theta], 1]] +
Sin[\[Theta]] Sin[Subscript[\[Theta], 1]]
Cos[\[Phi] - Subscript[\[Phi], 1]]] /; 0 < \[Theta] < Pi/2 &&
0 < Subscript[\[Theta], 1] < Pi/2 && 0 < \[Phi] < Pi/2 &&
0 < Subscript[\[Phi], 1] < Pi/2 && Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "n"]]]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]], RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", "2", ",", RowBox[List["Cos", "[", "\[Theta]", "]"]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", "2", ",", RowBox[List["Cos", "[", SubscriptBox["\[Theta]", "1"], "]"]]]], "]"]], RowBox[List["Cos", "[", RowBox[List["m", " ", RowBox[List["(", RowBox[List["\[Phi]", "-", SubscriptBox["\[Phi]", "1"]]], ")"]]]], "]"]]]]]], "\[Equal]", RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "\[Theta]", "]"]], " ", RowBox[List["Cos", "[", SubscriptBox["\[Theta]", "1"], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "\[Theta]", "]"]], " ", RowBox[List["Sin", "[", SubscriptBox["\[Theta]", "1"], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Phi]", "-", SubscriptBox["\[Phi]", "1"]]], "]"]]]]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Theta]", "<", RowBox[List["\[Pi]", "/", "2"]]]], "\[And]", RowBox[List["0", "<", SubscriptBox["\[Theta]", "1"], "<", RowBox[List["\[Pi]", "/", "2"]]]], "\[And]", RowBox[List["0", "<", "\[Phi]", "<", RowBox[List["\[Pi]", "/", "2"]]]], "\[And]", RowBox[List["0", "<", SubscriptBox["\[Phi]", "1"], "<", RowBox[List["\[Pi]", "/", "2"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ϕ </mi> <mo> - </mo> <msub> <mi> ϕ </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> θ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["cos", "(", "\[Theta]", ")"]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> θ </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["cos", "(", SubscriptBox["\[Theta]", "1"], ")"]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ⩵ </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> θ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> θ </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ϕ </mi> <mo> - </mo> <msub> <mi> ϕ </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> θ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> θ </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> θ </mi> <mo> < </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <msub> <mi> θ </mi> <mn> 1 </mn> </msub> <mo> < </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> ϕ </mi> <mo> < </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <msub> <mi> ϕ </mi> <mn> 1 </mn> </msub> <mo> < </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> m </ci> <apply> <plus /> <ci> ϕ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ϕ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> m </ci> <cn type='integer'> 2 </cn> <apply> <cos /> <ci> θ </ci> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> m </ci> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> θ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> θ </ci> </apply> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> θ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <cos /> <apply> <plus /> <ci> ϕ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ϕ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <ci> θ </ci> </apply> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> θ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> θ </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <ci> Subscript </ci> <ci> θ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> ϕ </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <ci> Subscript </ci> <ci> ϕ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m_", "=", RowBox[List["-", "n_"]]]], "n_"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_"]], ")"]], "!"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", "2", ",", RowBox[List["Cos", "[", "\[Theta]_", "]"]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", "2", ",", RowBox[List["Cos", "[", SubscriptBox["\[Theta]_", "1"], "]"]]]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["m_", " ", RowBox[List["(", RowBox[List["\[Phi]_", "-", SubscriptBox["\[Phi]_", "1"]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n_", "+", "m_"]], ")"]], "!"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "\[Theta]", "]"]], " ", RowBox[List["Cos", "[", SubscriptBox["\[Theta]\[Theta]", "1"], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "\[Theta]", "]"]], " ", RowBox[List["Sin", "[", SubscriptBox["\[Theta]\[Theta]", "1"], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Phi]", "-", SubscriptBox["\[Phi]\[Phi]", "1"]]], "]"]]]]]]]], "]"]], "/;", RowBox[List[RowBox[List["0", "<", "\[Theta]", "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["0", "<", SubscriptBox["\[Theta]\[Theta]", "1"], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["0", "<", "\[Phi]", "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["0", "<", SubscriptBox["\[Phi]\[Phi]", "1"], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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| LegendreP[n,z] | | LegendreP[nu,z] | | LegendreP[nu,mu,z] | | LegendreP[n,mu,2,z] | | LegendreP[nu,mu,3,z] | | |
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){kind=small}
