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http://functions.wolfram.com/07.12.25.0001.01
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Limit[LegendreQ[\[Nu], \[Mu], 3, Cosh[z/\[Nu]]]/\[Nu]^\[Mu],
\[Nu] -> Infinity] == E^(\[Mu] Pi I) BesselK[\[Mu], z]
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Cell[BoxData[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", RowBox[List["Cosh", "[", FractionBox["z", "\[Nu]"], "]"]]]], "]"]]]], ",", RowBox[List["\[Nu]", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["BesselK", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]]]]]]
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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> <munder> <mi> lim </mi> <mrow> <mi> ν </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msup> <mi> ν </mi> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mi> ν </mi> </mfrac> <mo> ) </mo> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <limit /> <bvar> <ci> ν </ci> </bvar> <condition> <apply> <tendsto /> <ci> ν </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <apply> <cosh /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> μ </ci> <pi /> <imaginaryi /> </apply> </apply> <apply> <ci> BesselK </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["\[Nu]_", RowBox[List["-", "\[Mu]_"]]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", RowBox[List["Cosh", "[", FractionBox["z_", "\[Nu]_"], "]"]]]], "]"]]]], ",", RowBox[List["\[Nu]_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
