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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.12.03.0020.01









  


  










Input Form





LegendreQ[3, \[Mu], 3, z] == (((Pi Csc[Pi \[Mu]])/(2 Gamma[4 - \[Mu]])) E^(I Pi \[Mu]) ((z + 1)^\[Mu] (15 z^3 + 4 \[Mu] - 15 z^2 \[Mu] - \[Mu]^3 + z (-9 + 6 \[Mu]^2)) - (z - 1)^\[Mu] (15 z^3 - 4 \[Mu] + 15 z^2 \[Mu] + \[Mu]^3 + z (-9 + 6 \[Mu]^2))))/((z - 1)^(\[Mu]/2) (z + 1)^(\[Mu]/2))










Standard Form





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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> 3 </mn> <mi> &#956; </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <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> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <msup> <mi> &#956; </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> &#956; </mi> <mn> 3 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </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'> 3 </cn> </apply> <ci> &#956; </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <csc /> <apply> <times /> <pi /> <ci> &#956; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#956; </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; 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</ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#956; </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <ci> &#956; </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -9 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> &#956; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </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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