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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.09.03.0016.01









  


  










Input Form





LegendreP[5, \[Mu], 3, z] == (1/Gamma[6 - \[Mu]]) (945 z^5 - 64 \[Mu] - 945 z^4 \[Mu] + 20 \[Mu]^3 - \[Mu]^5 - 105 z^2 \[Mu] (-7 + \[Mu]^2) + 210 z^3 (-5 + 2 \[Mu]^2) + 15 z (15 - 13 \[Mu]^2 + \[Mu]^4)) ((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["5", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["6", "-", "\[Mu]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List["945", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["64", " ", "\[Mu]"]], "-", RowBox[List["945", " ", SuperscriptBox["z", "4"], " ", "\[Mu]"]], "+", RowBox[List["20", " ", SuperscriptBox["\[Mu]", "3"]]], "-", SuperscriptBox["\[Mu]", "5"], "-", RowBox[List["105", " ", SuperscriptBox["z", "2"], " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "+", RowBox[List["210", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]], "+", RowBox[List["15", " ", "z", " ", RowBox[List["(", RowBox[List["15", "-", RowBox[List["13", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"]]], ")"]]]]]], ")"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "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> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ] </annotation> </semantics> <mn> 5 </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> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 945 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mrow> <mn> 945 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> &#956; </mi> <mn> 5 </mn> </msup> <mo> - </mo> <mrow> <mn> 105 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> + </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 210 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 15 </mn> <mo> - </mo> <mrow> <mn> 13 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> <cn type='integer'> 5 </cn> </apply> <ci> &#956; </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 64 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> -7 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 210 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <cn type='integer'> -5 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <ci> z </ci> <apply> <plus /> <cn type='integer'> 15 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["5", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["945", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["64", " ", "\[Mu]"]], "-", RowBox[List["945", " ", SuperscriptBox["z", "4"], " ", "\[Mu]"]], "+", RowBox[List["20", " ", SuperscriptBox["\[Mu]", "3"]]], "-", SuperscriptBox["\[Mu]", "5"], "-", RowBox[List["105", " ", SuperscriptBox["z", "2"], " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "+", RowBox[List["210", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]], "+", RowBox[List["15", " ", "z", " ", RowBox[List["(", RowBox[List["15", "-", RowBox[List["13", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["6", "-", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]]]]]]










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





2001-10-29





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