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http://functions.wolfram.com/07.09.26.0081.01
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LegendreP[\[Nu], \[Mu], 3, z] ==
((-Pi) Csc[Pi \[Mu]] ((1/(Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]]))
Hypergeometric2F1Regularized[-\[Nu], 1 + \[Nu], 1 + \[Mu], (1 + z)/2] +
(((2^\[Mu] Sin[Pi \[Nu]])/Pi) Hypergeometric2F1Regularized[
1 - \[Mu] + \[Nu], -\[Mu] - \[Nu], 1 - \[Mu], (1 + z)/2])/
(1 + z)^\[Mu]))/((-1 + z)/(1 + z))^(\[Mu]/2) /;
!Element[\[Mu], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], RowBox[List["1", "+", "z"]]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "\[Pi]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]]]]]]
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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> <msubsup> <semantics> <mi> 𝔓 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreP] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> μ </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> μ </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> μ </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <ci> 𝔓 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> μ </ci> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> μ </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], RowBox[List["1", "+", "z"]]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], "\[Pi]"]]], ")"]]]], "/;", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]] |
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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,2,z] | | |
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){kind=small}
