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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,2,z] > Series representations > Asymptotic series expansions > Expansions at nu==-1/2+i infinity





http://functions.wolfram.com/07.08.06.0058.01









  


  








Input Form





LegendreP[-(1/2) + I \[Tau], \[Mu], 2, 3] \[Proportional] Sqrt[2/Pi] \[Tau]^\[Mu] 2^(I \[Tau] - 1/2) BesselJ[-\[Mu], \[Tau] Sqrt[2]] (1 + O[1/\[Tau]]) /; (\[Tau] -> Infinity) && \[Mu] < 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Tau]"]]]], ",", "\[Mu]", ",", "2", ",", "3"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], SuperscriptBox["\[Tau]", "\[Mu]"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Tau]"]], "-", FractionBox["1", "2"]]]], RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], ",", RowBox[List["\[Tau]", " ", SqrtBox["2"]]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "\[Tau]"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Tau]", "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Mu]", "<", "0"]]]]]]]]








MathML Form







<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[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mn> 3 </mn> <annotation encoding='Mathematica'> TagBox[&quot;3&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#964; </mi> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> &#964; </mi> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> &#964; </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#964; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#956; </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LegendreP </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> &#964; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> &#964; </ci> <ci> &#956; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> &#964; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <ci> &#964; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> &#964; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> &#964; </ci> <infinity /> </apply> <apply> <lt /> <ci> &#956; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Tau]_"]]]], ",", "\[Mu]_", ",", "2", ",", "3"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox["\[Tau]", "\[Mu]"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Tau]"]], "-", FractionBox["1", "2"]]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], ",", RowBox[List["\[Tau]", " ", SqrtBox["2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "\[Tau]"], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Tau]", "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["\[Mu]", "<", "0"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2007-05-02