Assosiated Legendre function of the second kind of type 2: Specific values (formula 07.11.03.0023)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

http://functions.wolfram.com/07.11.03.0023.01

Input Form

LegendreQ[9, \[Mu], 2, z] == (((Pi Csc[Pi \[Mu]])/(2 Gamma[10 - \[Mu]])) ((-(1 - z)^\[Mu]) (34459425 z^9 + 34459425 z^8 \[Mu] + (-8 + \[Mu]) (-6 + \[Mu]) (-4 + \[Mu]) (-2 + \[Mu]) \[Mu] (2 + \[Mu]) (4 + \[Mu]) (6 + \[Mu]) (8 + \[Mu]) + 4729725 z^6 \[Mu] (-13 + \[Mu]^2) + 8108100 z^7 (-9 + 2 \[Mu]^2) + 135135 z^4 \[Mu] (249 - 40 \[Mu]^2 + \[Mu]^4) + 945945 z^5 (54 - 25 \[Mu]^2 + \[Mu]^4) + 495 z^2 \[Mu] (-11601 + 2933 \[Mu]^2 - 154 \[Mu]^4 + 2 \[Mu]^6) + 6930 z^3 (-1890 + 1373 \[Mu]^2 - 115 \[Mu]^4 + 2 \[Mu]^6) + 45 z (19845 - 20217 \[Mu]^2 + 2674 \[Mu]^4 - 98 \[Mu]^6 + \[Mu]^8)) + (1 + z)^\[Mu] (34459425 z^9 - 34459425 z^8 \[Mu] - (-8 + \[Mu]) (-6 + \[Mu]) (-4 + \[Mu]) (-2 + \[Mu]) \[Mu] (2 + \[Mu]) (4 + \[Mu]) (6 + \[Mu]) (8 + \[Mu]) - 4729725 z^6 \[Mu] (-13 + \[Mu]^2) + 8108100 z^7 (-9 + 2 \[Mu]^2) - 135135 z^4 \[Mu] (249 - 40 \[Mu]^2 + \[Mu]^4) + 945945 z^5 (54 - 25 \[Mu]^2 + \[Mu]^4) - 495 z^2 \[Mu] (-11601 + 2933 \[Mu]^2 - 154 \[Mu]^4 + 2 \[Mu]^6) + 6930 z^3 (-1890 + 1373 \[Mu]^2 - 115 \[Mu]^4 + 2 \[Mu]^6) + 45 z (19845 - 20217 \[Mu]^2 + 2674 \[Mu]^4 - 98 \[Mu]^6 + \[Mu]^8)) Cos[Pi \[Mu]]))/(1 - z^2)^(\[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> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mn> 9 </mn> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </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> μ </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 34459425 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 34459425 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8108100 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4729725 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 13 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 945945 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 25 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 54 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 135135 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 40 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 249 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6930 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 115 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1373 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1890 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 495 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 154 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2933 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 11601 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 8 </mn> </msup> <mo> - </mo> <mrow> <mn> 98 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2674 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 20217 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 19845 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> μ </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> μ </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 34459425 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 34459425 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8108100 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4729725 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 13 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 945945 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 25 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 54 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 135135 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 40 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 249 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6930 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 115 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1373 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1890 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 495 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 154 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2933 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 11601 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 8 </mn> </msup> <mo> - </mo> <mrow> <mn> 98 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2674 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 20217 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 19845 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( 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Rule Form

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

2001-10-29

LegendreQ[nu,z]
LegendreQ[nu,mu,z]
LegendreQ[nu,mu,3,z]