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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Specific values > Specialized values > For fixed z





http://functions.wolfram.com/07.11.03.0138.01









  


  










Input Form





LegendreQ[9, 8, 2, z] == (-(45/(2 (-1 + z^2)^4))) (2 (229376 + 17 z^2 (-184331 + z^2 (801535 + 13 z^2 (-140903 + 11 z^2 (17609 - 15159 z^2 + 8043 z^4 - 2415 z^6 + 315 z^8))))) + 765765 z (-1 + z^2)^8 (Log[1 - z] - Log[1 + z]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["9", ",", "8", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["45", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "4"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["229376", "+", RowBox[List["17", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "184331"]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["801535", "+", RowBox[List["13", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "140903"]], "+", RowBox[List["11", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["17609", "-", RowBox[List["15159", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8043", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["2415", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["315", " ", SuperscriptBox["z", "8"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["765765", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]], ")"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mn> 9 </mn> <mn> 8 </mn> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 45 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 765765 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 17 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 13 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 11 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 315 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2415 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8043 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15159 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 17609 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 140903 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 801535 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 184331 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 229376 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreQ </ci> <cn type='integer'> 9 </cn> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 765765 </cn> <ci> z </ci> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 17 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2415 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8043 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15159 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 17609 </cn> </apply> </apply> <cn type='integer'> -140903 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 801535 </cn> </apply> </apply> <cn type='integer'> -184331 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 229376 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["9", ",", "8", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["229376", "+", RowBox[List["17", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "184331"]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["801535", "+", RowBox[List["13", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "140903"]], "+", RowBox[List["11", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["17609", "-", RowBox[List["15159", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8043", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["2415", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["315", " ", SuperscriptBox["z", "8"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["765765", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "4"]]]]]]]]]]










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





2001-10-29





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