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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/07.09.03.0019.01









  


  










Input Form





LegendreP[8, \[Mu], 3, z] == (1/Gamma[9 - \[Mu]]) (11025 + 2027025 z^8 - 2027025 z^7 \[Mu] - 12916 \[Mu]^2 + 1974 \[Mu]^4 - 84 \[Mu]^6 + \[Mu]^8 + 945945 z^6 (-4 + \[Mu]^2) - 135135 z^5 \[Mu] (-23 + 2 \[Mu]^2) + 51975 z^4 (42 - 22 \[Mu]^2 + \[Mu]^4) - 3465 z^3 \[Mu] (383 - 70 \[Mu]^2 + 2 \[Mu]^4) + 315 z^2 (-1260 + 1043 \[Mu]^2 - 100 \[Mu]^4 + 2 \[Mu]^6) - 9 z \[Mu] (-15159 + 4396 \[Mu]^2 - 266 \[Mu]^4 + 4 \[Mu]^6)) ((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["8", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", " ", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["9", "-", "\[Mu]"]], "]"]]], RowBox[List["(", RowBox[List["11025", "+", RowBox[List["2027025", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["2027025", " ", SuperscriptBox["z", "7"], " ", "\[Mu]"]], "-", RowBox[List["12916", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["1974", " ", SuperscriptBox["\[Mu]", "4"]]], "-", RowBox[List["84", " ", SuperscriptBox["\[Mu]", "6"]]], "+", SuperscriptBox["\[Mu]", "8"], "+", RowBox[List["945945", " ", SuperscriptBox["z", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "-", RowBox[List["135135", " ", SuperscriptBox["z", "5"], " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "23"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]], "+", RowBox[List["51975", " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["42", "-", RowBox[List["22", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"]]], ")"]]]], "-", RowBox[List["3465", " ", SuperscriptBox["z", "3"], " ", "\[Mu]", " ", RowBox[List["(", RowBox[List["383", "-", RowBox[List["70", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]]]], "+", RowBox[List["315", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1260"]], "+", RowBox[List["1043", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["100", " ", SuperscriptBox["\[Mu]", "4"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "6"]]]]], ")"]]]], "-", RowBox[List["9", " ", "z", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "15159"]], "+", RowBox[List["4396", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["266", " ", SuperscriptBox["\[Mu]", "4"]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "6"]]]]], ")"]]]]]], ")"]], 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> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mn> 8 </mn> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 11025 </mn> <mo> + </mo> <mrow> <mn> 2027025 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2027025 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mrow> <mn> 12916 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1974 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 84 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#956; </mi> <mn> 8 </mn> </msup> <mo> + </mo> <mrow> <mn> 945945 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> + </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 135135 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 23 </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> 51975 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 42 </mn> <mo> - </mo> <mrow> <mn> 22 </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> <mo> - </mo> <mrow> <mn> 3465 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 383 </mn> <mo> - </mo> <mrow> <mn> 70 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 315 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1260 </mn> </mrow> <mo> + </mo> <mrow> <mn> 1043 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 100 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 6 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 15159 </mn> </mrow> <mo> + </mo> <mrow> <mn> 4396 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 266 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 6 </mn> </msup> </mrow> </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> <ci> LegendreP </ci> <cn type='integer'> 8 </cn> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 11025 </cn> <apply> <times /> <cn type='integer'> 2027025 </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'> 2027025 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12916 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1974 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 84 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> 945945 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <cn type='integer'> -4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 135135 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> -23 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 51975 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <cn type='integer'> 42 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 22 </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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3465 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> 383 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 70 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> -1260 </cn> <apply> <times /> <cn type='integer'> 1043 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 100 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> -15159 </cn> <apply> <times /> <cn type='integer'> 4396 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 266 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 6 </cn> </apply> </apply> </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["8", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["11025", "+", RowBox[List["2027025", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["2027025", " ", SuperscriptBox["z", "7"], " ", "\[Mu]"]], "-", RowBox[List["12916", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["1974", " ", SuperscriptBox["\[Mu]", "4"]]], "-", RowBox[List["84", " ", SuperscriptBox["\[Mu]", "6"]]], "+", SuperscriptBox["\[Mu]", "8"], "+", RowBox[List["945945", " ", SuperscriptBox["z", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "-", RowBox[List["135135", " ", SuperscriptBox["z", "5"], " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "23"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]], "+", RowBox[List["51975", " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["42", "-", RowBox[List["22", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"]]], ")"]]]], "-", RowBox[List["3465", " ", SuperscriptBox["z", "3"], " ", "\[Mu]", " ", RowBox[List["(", RowBox[List["383", "-", RowBox[List["70", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]]]], "+", RowBox[List["315", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1260"]], "+", RowBox[List["1043", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["100", " ", SuperscriptBox["\[Mu]", "4"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "6"]]]]], ")"]]]], "-", RowBox[List["9", " ", "z", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "15159"]], "+", RowBox[List["4396", " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["266", " ", SuperscriptBox["\[Mu]", "4"]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "6"]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["9", "-", "\[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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