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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/07.08.06.0057.01









  


  








Input Form





LegendreP[-(1/2) + I \[Tau], \[Mu], 2, x] \[Proportional] Sqrt[2/Pi] ((x + 1)/(x - 1))^(\[Mu]/2) (\[Tau]^(\[Mu] - 1/2)/Sqrt[x - 1]) Cos[\[Tau] Log[2 (x - 1)] + (\[Mu] - 1/2) (Pi/2) + \[Tau]/(x - 1)] (1 + O[1/\[Tau]]) /; (\[Tau] -> Infinity) && Element[x, Reals] && x > 3








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", ",", "x"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "+", "1"]], RowBox[List["x", "-", "1"]]], ")"]], FractionBox["\[Mu]", "2"]], FractionBox[SuperscriptBox["\[Tau]", RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]]], SqrtBox[RowBox[List["x", "-", "1"]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Tau]", " ", RowBox[List["Log", "[", RowBox[List["2", RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]], ")"]], FractionBox["\[Pi]", "2"]]], "+", FractionBox["\[Tau]", RowBox[List["x", "-", "1"]]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "\[Tau]"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Tau]", "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "3"]]]]]]]]








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> <mi> x </mi> <annotation encoding='Mathematica'> TagBox[&quot;x&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> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#964; </mi> <mrow> <mi> &#956; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mi> &#964; </mi> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> &#964; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </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> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &gt; </mo> <mn> 3 </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> <ci> x </ci> </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> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#964; </ci> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <pi /> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <ci> &#964; </ci> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#964; </ci> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </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> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 3 </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", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "+", "1"]], RowBox[List["x", "-", "1"]]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", SuperscriptBox["\[Tau]", RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Tau]", " ", RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]], ")"]], " ", "\[Pi]"]], "+", FractionBox["\[Tau]", RowBox[List["x", "-", "1"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "\[Tau]"], "]"]]]], ")"]]]], SqrtBox[RowBox[List["x", "-", "1"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Tau]", "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "3"]]]]]]]]]]


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





2007-05-02