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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > General characteristics > Branch cuts > With respect to z





http://functions.wolfram.com/07.11.04.0023.01









  


  










Input Form





Limit[LegendreQ[\[Nu], \[Mu], 2, x - I \[Epsilon]], \[Epsilon] -> Plus[0]] == (-(1/2)) I Pi (3 Cos[Pi \[Nu]] + Cos[Pi (2 \[Mu] + \[Nu])]) LegendreP[\[Nu], \[Mu], 2, -x] + 2 I Cos[Pi \[Mu]] Sin[Pi (\[Mu] + \[Nu])] LegendreQ[\[Nu], \[Mu], 2, -x] + I Pi Cos[Pi \[Mu]] LegendreP[\[Nu], \[Mu], 2, x] + LegendreQ[\[Nu], \[Mu], 2, x]/E^(I Pi \[Mu]) /; x < -1










Standard Form





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MathML Form







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</mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <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> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> x </mi> <annotation encoding='Mathematica'> TagBox[&quot;x&quot;, HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, &quot;x&quot;]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> &#1013; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> &#1013; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <cos /> <apply> <times /> <pi /> <ci> &#956; </ci> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <ci> &#956; </ci> </apply> </apply> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <cos /> <apply> <times /> <pi /> <ci> &#956; </ci> </apply> </apply> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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