Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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] > Specific values > Specialized values > For fixed mu, z





http://functions.wolfram.com/07.11.03.0022.01









  


  








Input Form





LegendreQ[8, \[Mu], 2, z] == (((Pi Csc[Pi \[Mu]])/(2 Gamma[9 - \[Mu]])) ((1 + z)^\[Mu] (2027025 z^8 - 2027025 z^7 \[Mu] + (-7 + \[Mu]) (-5 + \[Mu]) (-3 + \[Mu]) (-1 + \[Mu]) (1 + \[Mu]) (3 + \[Mu]) (5 + \[Mu]) (7 + \[Mu]) + 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 (-6 + \[Mu]) (6 + \[Mu]) (35 - 28 \[Mu]^2 + 2 \[Mu]^4) - 9 z \[Mu] (-15159 + 4396 \[Mu]^2 - 266 \[Mu]^4 + 4 \[Mu]^6)) Cos[Pi \[Mu]] - (1 - z)^\[Mu] (2027025 z^8 + 2027025 z^7 \[Mu] + (-7 + \[Mu]) (-5 + \[Mu]) (-3 + \[Mu]) (-1 + \[Mu]) (1 + \[Mu]) (3 + \[Mu]) (5 + \[Mu]) (7 + \[Mu]) + 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 (-6 + \[Mu]) (6 + \[Mu]) (35 - 28 \[Mu]^2 + 2 \[Mu]^4) + 9 z \[Mu] (-15159 + 4396 \[Mu]^2 - 266 \[Mu]^4 + 4 \[Mu]^6)) Csc[Pi \[Mu]]))/(1 - z^2)^(\[Mu]/2)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["8", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["9", "-", "\[Mu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2027025", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["2027025", " ", SuperscriptBox["z", "7"], " ", "\[Mu]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["7", "+", "\[Mu]"]], ")"]]]], "+", 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["-", "6"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["28", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]]]], "-", 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"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2027025", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["2027025", " ", SuperscriptBox["z", "7"], " ", "\[Mu]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["7", "+", "\[Mu]"]], ")"]]]], "+", 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["-", "6"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["28", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]]]], "+", 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"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]]]], ")"]]]]]]]]








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> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mn> 8 </mn> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </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> 9 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </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> 2027025 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2027025 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 945945 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 135135 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 23 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 51975 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 22 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 42 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3465 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 70 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 383 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 315 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 28 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 35 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 6 </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> 4396 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 15159 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <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> <mi> &#956; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 945945 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 135135 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 23 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 51975 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 22 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 42 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3465 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 70 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 383 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 315 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 28 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 35 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 6 </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> 4396 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 15159 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreQ </ci> <cn type='integer'> 8 </cn> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <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'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <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'> 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> <ci> &#956; </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 945945 </cn> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 135135 </cn> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -23 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 51975 </cn> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> <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> <cn type='integer'> 42 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3465 </cn> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </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'> 70 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 383 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -6 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </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'> 28 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 35 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 6 </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'> 4396 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -15159 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -7 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -5 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2027025 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2027025 </cn> <ci> &#956; </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 945945 </cn> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 135135 </cn> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -23 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 51975 </cn> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> <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> <cn type='integer'> 42 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3465 </cn> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </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'> 70 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 383 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -6 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </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'> 28 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 35 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <ci> &#956; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 6 </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'> 4396 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -15159 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -7 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -5 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> &#956; </ci> </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["8", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2027025", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["2027025", " ", SuperscriptBox["z", "7"], " ", "\[Mu]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["7", "+", "\[Mu]"]], ")"]]]], "+", 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["-", "6"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["28", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]]]], "-", 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"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2027025", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["2027025", " ", SuperscriptBox["z", "7"], " ", "\[Mu]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["7", "+", "\[Mu]"]], ")"]]]], "+", 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["-", "6"]], "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["28", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "4"]]]]], ")"]]]], "+", 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"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["9", "-", "\[Mu]"]], "]"]]]]]]]]]


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





2001-10-29