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http://functions.wolfram.com/07.09.27.0004.01
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LegendreP[\[Nu], \[Mu], 3, x] == E^((Pi I \[Mu])/2)
Limit[LegendreP[\[Nu], \[Mu], 2, z], z -> x, Direction -> I] /; x > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "x"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]], "2"]], RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], ",", RowBox[List["z", "\[Rule]", "x"]], ",", RowBox[List["Direction", "\[Rule]", "\[ImaginaryI]"]]]], "]"]]]]]], "/;", RowBox[List["x", ">", "1"]]]]]]
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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]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", 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> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔓 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <limit /> <bvar> <ci> z </ci> </bvar> <condition> <apply> <tendsto /> <ci> z </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </condition> <apply> <ci> LegendreP </ci> <ci> ν </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </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", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]], "2"]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], ",", RowBox[List["z", "\[Rule]", "x"]], ",", RowBox[List["Direction", "\[Rule]", "\[ImaginaryI]"]]]], "]"]]]], "/;", RowBox[List["x", ">", "1"]]]]]]]] |
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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}
