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 LegendreP

 http://functions.wolfram.com/07.08.06.0052.01

 Input Form

 LegendreP[\[Nu], \[Mu], 2, z] == ((Sin[Pi \[Nu]]/(Pi Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]])) (x + 1)^((1/2) \[Mu]) Exp[Pi I \[Mu] Floor[Arg[z - x]/(2 Pi)]] ((-2 Pi^2 I Floor[Arg[-x + z]/(2 Pi)] Csc[Pi \[Nu]] Hypergeometric2F1Regularized[-\[Nu], 1 + \[Nu], 1 + \[Mu], (1 + x)/2])/ E^(I Pi \[Mu] Floor[Arg[-x + z]/(2 Pi)]) - MeijerG[{{1 + \[Nu], -\[Nu]}, {}}, {{0, -\[Mu]}, {}}, (1 + x)/2]/ E^(2 I Pi \[Mu] Floor[Arg[-x + z]/(2 Pi)]) + ((1/(2 (-1 + x^2))) (2 Pi^2 I E^(I Pi \[Mu] Floor[Arg[-x + z]/(2 Pi)]) Floor[Arg[-x + z]/(2 Pi)] Csc[Pi \[Nu]] ((-1 + x^2) \[Nu] Hypergeometric2F1Regularized[1 - \[Nu], 2 + \[Nu], 2 + \[Mu], (1 + x)/2] + 2 \[Mu] (1 + \[Nu]) Hypergeometric2F1Regularized[-\[Nu], 1 + \[Nu], 1 + \[Mu], (1 + x)/2]) + (-1 + x^2) MeijerG[{{\[Nu], -1 - \[Nu]}, {}}, {{0, -1 - \[Mu]}, {}}, (1 + x)/2] + 2 \[Mu] MeijerG[{{1 + \[Nu], -\[Nu]}, {}}, {{0, -\[Mu]}, {}}, (1 + x)/2]) (z - x))/E^(2 I Pi \[Mu] Floor[Arg[-x + z]/(2 Pi)]) + \[Ellipsis]))/(1 - x)^((1/2) \[Mu]) /; (z -> x) && Element[x, Reals] && x < -1

 Standard Form

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 MathML Form

 P TagBox["P", LegendreP] ν μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) sin ( π ν ) π Γ ( - μ - ν ) Γ ( - μ + ν + 1 ) ( 1 - x ) - μ 2 ( x + 1 ) μ 2 π μ arg ( z - x ) 2 π ( 2 π 2 - π μ arg ( z - x ) 2 π arg ( z - x ) 2 π csc ( π ν ) 2 F ~ 1 ( - ν , ν + 1 ; μ + 1 ; x + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - - 2 π μ arg ( z - x ) 2 π G 2 , 2 2 , 2 ( x + 1 2 ν + 1 , - ν 0 , - μ ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + 1 2 ( x 2 - 1 ) - 2 π μ arg ( z - x ) 2 π ( 2 π μ arg ( z - x ) 2 π π 2 csc ( π ν ) arg ( z - x ) 2 π ( ( x 2 - 1 ) ν 2 F ~ 1 ( 1 - ν , ν + 2 ; μ + 2 ; x + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + 2 μ ( ν + 1 ) 2 F ~ 1 ( - ν , ν + 1 ; μ + 1 ; x + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) + ( x 2 - 1 ) G 2 , 2 2 , 2 ( x + 1 2 ν , - ν - 1 0 , - μ - 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["\[Nu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + 2 μ G 2 , 2 2 , 2 ( x + 1 2 ν + 1 , - ν 0 , - μ ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ( z - x ) + ) /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < - 1 Condition LegendreP ν μ 2 z ν Gamma -1 μ -1 ν Gamma -1 μ ν 1 -1 1 -1 x -1 μ 2 -1 x 1 μ 2 -1 μ z -1 x 2 -1 2 2 -1 μ z -1 x 2 -1 z -1 x 2 -1 ν Hypergeometric2F1Regularized -1 ν ν 1 μ 1 x 1 2 -1 -1 -2 μ z -1 x 2 -1 MeijerG ν 1 -1 ν 0 -1 μ x 1 2 -1 1 2 x 2 -1 -1 -2 μ z -1 x 2 -1 2 μ z -1 x 2 -1 2 ν z -1 x 2 -1 x 2 -1 ν Hypergeometric2F1Regularized 1 -1 ν ν 2 μ 2 x 1 2 -1 2 μ ν 1 Hypergeometric2F1Regularized -1 ν ν 1 μ 1 x 1 2 -1 x 2 -1 MeijerG ν -1 ν -1 0 -1 μ -1 x 1 2 -1 2 μ MeijerG ν 1 -1 ν 0 -1 μ x 1 2 -1 z -1 x Rule z x x x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", "1"]], ")"]], FractionBox["\[Mu]", "2"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Mu]", " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02