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 | | http://functions.wolfram.com/07.09.26.0041.01 | 
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 | | LegendreP[\[Nu], -\[Mu], 3, Sqrt[z + 1]] LegendreQ[\[Nu], \[Mu], 3, 
    Sqrt[z + 1]] == (E^(Pi I \[Mu])/(2 Sqrt[Pi])) 
   MeijerG[{{1/2, -\[Nu]}, {\[Nu] + 1}}, {{0, \[Mu]}, {-\[Mu]}}, z] /; 
  !IntervalMemberQ[Interval[{-Infinity, -1}], z] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", SqrtBox[RowBox[List["z", "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[RowBox[List["z", "+", "1"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Mu]"]], "}"]], ",", RowBox[List["{", RowBox[List["-", "\[Mu]"]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]] | 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <mrow>  <mrow>  <msubsup>  <semantics>  <mi> 𝔓 </mi>  <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreP] </annotation>  </semantics>  <mi> ν </mi>  <mrow>  <mo> - </mo>  <mi> μ </mi>  </mrow>  </msubsup>  <mo> ( </mo>  <semantics>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List["z", "+", "1"]]], HoldComplete[LegendreP, 3]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <semantics>  <mi> 𝔔 </mi>  <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation>  </semantics>  <mi> ν </mi>  <mi> μ </mi>  </msubsup>  <mo> ( </mo>  <semantics>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List["z", "+", "1"]]], HoldComplete[LegendreQ, 3]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> μ </mi>  </mrow>  </msup>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msqrt>  <mi> π </mi>  </msqrt>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 3 </mn>  <mo> , </mo>  <mn> 3 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mi> ν </mi>  </mrow>  <mo> , </mo>  <mrow>  <mi> ν </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mi> μ </mi>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mi> μ </mi>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["\[Mu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; 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</mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", RowBox[List["-", "\[Mu]_"]], ",", "3", ",", SqrtBox[RowBox[List["z_", "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[RowBox[List["z_", "+", "1"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Mu]"]], "}"]], ",", RowBox[List["{", RowBox[List["-", "\[Mu]"]], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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 | | LegendreP[n,z] |  | LegendreP[nu,z] |  | LegendreP[nu,mu,z] |  | LegendreP[n,mu,2,z] |  | LegendreP[nu,mu,2,z] |  |  | 
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