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variants of this functions
LegendreQ




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,z] > Representations through more general functions > Through Meijer G > Classical cases for the direct function itself





http://functions.wolfram.com/07.10.26.0008.01









  


  








Input Form





LegendreQ[\[Nu], z] == (Tan[Pi \[Nu]]/(4 Pi)) (Log[-z - 1] - Log[1 + z]) MeijerG[{{1, 1}, {}}, {{-\[Nu]}, {1 + \[Nu]}}, 2/(z - 1)] + (1/4) (2 - (Tan[Pi \[Nu]]/Pi) (Log[-z - 1] - Log[1 + z])) MeijerG[{{1, 1}, {}}, {{1 + \[Nu]}, {-\[Nu]}}, 2/(z - 1)] /; !Element[\[Nu], Integers]








Standard Form





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MathML Form







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</mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&quot;]], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;2&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[&quot;2&quot;, RowBox[List[&quot;z&quot;, &quot;-&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[&quot;1&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> tan </mi> <mo> &#8290; 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</ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </list> </list> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> tan </ci> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </list> <list /> </list> <list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </list> <list> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </list> </list> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> &#957; </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox["2", RowBox[List["z", "-", "1"]]]]], "]"]]]], RowBox[List["4", " ", "\[Pi]"]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List[RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]], ")"]]]], "\[Pi]"]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List["-", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox["2", RowBox[List["z", "-", "1"]]]]], "]"]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]


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





2001-10-29