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,2,z] > Representations through more general functions > Through Meijer G > Classical cases involving unit step theta





http://functions.wolfram.com/07.11.26.0019.01









  


  










Input Form





(UnitStep[1 - Abs[z]]/Sqrt[1 - z]) (LegendreQ[\[Nu], \[Mu], 2, -Sqrt[1 - z]] + LegendreQ[\[Nu], \[Mu], 2, Sqrt[1 - z]]) == (-((2^\[Mu] Pi^2 Tan[((\[Mu] + \[Nu])/2) Pi])/(Gamma[(1 - \[Mu] - \[Nu])/2] Gamma[1 + (\[Nu] - \[Mu])/2]))) MeijerG[{{}, {-(\[Nu]/2), (1 + \[Nu])/2}}, {{-(\[Mu]/2), \[Mu]/2}, {}}, z] /; !Element[\[Nu], Integers] && !Element[\[Mu], Integers] && !IntervalMemberQ[Interval[{-1, 0}], z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List["1", "-", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", RowBox[List["-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], "]"]], "+", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", SqrtBox[RowBox[List["1", "-", "z"]]]]], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", "\[Mu]"], SuperscriptBox["\[Pi]", "2"], RowBox[List["Tan", "[", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], "\[Pi]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"]]], " ", "]"]]]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "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> <mrow> <mfrac> <mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <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> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]]]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> + </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> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </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;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#956; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <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> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]]]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> + </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> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </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;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#956; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z_", "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", RowBox[List["-", SqrtBox[RowBox[List["1", "-", "z_"]]]]]]], "]"]], "+", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", SqrtBox[RowBox[List["1", "-", "z_"]]]]], "]"]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", "z_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Tan", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]], " ", "\[Pi]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.