Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Specific values > Specialized values > For fixed z





http://functions.wolfram.com/07.12.03.0147.01









  


  










Input Form





LegendreQ[10, 3, 3, z] == -(2 (49152 + 13 z^2 (-172353 + 11 z^2 (124885 + 3 z^2 (-128106 + 7 z^2 (26514 + 85 z^2 (-215 + 57 z^2)))))) - 45045 z (-1 + z^2)^3 (-7 + 105 z^2 - 357 z^4 + 323 z^6) (Log[z + 1] - Log[z - 1]))/(224 (z - 1)^(3/2) (z + 1)^(3/2))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["10", ",", "3", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["49152", "+", RowBox[List["13", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "172353"]], "+", RowBox[List["11", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["124885", "+", RowBox[List["3", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "128106"]], "+", RowBox[List["7", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["26514", "+", RowBox[List["85", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "215"]], "+", RowBox[List["57", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["45045", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["105", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["357", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["323", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["224", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]]]]]]










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> &#120084; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalQ]&quot;, LegendreQ] </annotation> </semantics> <mn> 10 </mn> <mn> 3 </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> 1 </mn> <mrow> <mn> 224 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <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> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 85 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 57 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 215 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 26514 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 128106 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 124885 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 172353 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 49152 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 45045 </mn> <mo> &#8290; </mo> <mi> z </mi> <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> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 323 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 357 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 105 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </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> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> &#120084; </ci> </apply> <cn type='integer'> 10 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 224 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <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'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 85 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 57 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -215 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 26514 </cn> </apply> </apply> <cn type='integer'> -128106 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 124885 </cn> </apply> </apply> <cn type='integer'> -172353 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 49152 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 45045 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 323 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 357 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -7 </cn> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </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> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["10", ",", "3", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["49152", "+", RowBox[List["13", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "172353"]], "+", RowBox[List["11", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["124885", "+", RowBox[List["3", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "128106"]], "+", RowBox[List["7", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["26514", "+", RowBox[List["85", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "215"]], "+", RowBox[List["57", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["45045", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["105", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["357", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["323", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]], RowBox[List["224", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.