html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 LegendreP

 http://functions.wolfram.com/07.08.06.0043.01

 Input Form

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

 Standard Form

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")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Mu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", "\[Mu]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", 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RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]", " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "0", "2"], "-", "1"]], ")"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "\[Nu]"]], ",", RowBox[List["2", "+", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]]]]]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], " ", "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]]]]]]

 MathML Form

 P TagBox["P", LegendreP] ν μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) sin ( π ν ) π Γ ( - μ - ν ) Γ ( 1 - μ + ν ) ( 1 1 - z 0 ) - 1 2 μ arg ( z 0 - z ) 2 π ( 1 - z 0 ) - 1 2 μ ( arg ( z 0 - z ) 2 π + 1 ) ( 1 z 0 + 1 ) 1 2 μ arg ( z - z 0 ) 2 π ( z 0 + 1 ) 1 2 μ ( arg ( z - z 0 ) 2 π + 1 ) ( 2 π - π μ arg ( z - z 0 ) 2 π arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π Γ ( - ν ) Γ ( ν + 1 ) 2 F ~ 1 ( - ν , ν + 1 ; μ + 1 ; z 0 + 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[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - ( 1 z 0 + 1 ) - μ arg ( z - z 0 ) 2 π ( z 0 + 1 ) - μ arg ( z - z 0 ) 2 π G 2 , 2 2 , 2 ( 1 2 ( z 0 + 1 ) 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[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "+", "1"]], ")"]]]], 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 ( z 0 2 - 1 ) - π μ arg ( z - z 0 ) 2 π ( 1 z 0 + 1 ) - μ arg ( z - z 0 ) 2 π ( z 0 + 1 ) - μ arg ( z - z 0 ) 2 π ( 2 π arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π ( 1 z 0 + 1 ) μ arg ( z - z 0 ) 2 π ( z 0 + 1 ) μ arg ( z - z 0 ) 2 π ( Γ ( 1 - ν ) Γ ( ν + 1 ) ( z 0 2 - 1 ) 2 F ~ 1 ( 1 - ν , ν + 2 ; μ + 2 ; z 0 + 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[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - 2 μ Γ ( - ν ) Γ ( ν + 2 ) 2 F ~ 1 ( - ν , ν + 1 ; μ + 1 ; z 0 + 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[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) + 2 π μ arg ( z - z 0 ) 2 π μ G 2 , 2 2 , 2 ( z 0 + 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[SubscriptBox["z", "0"], "+", "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 0 2 - 1 ) π μ arg ( z - z 0 ) 2 π G 2 , 2 2 , 2 ( z 0 + 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[SubscriptBox["z", "0"], "+", "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]] ) ( z - z 0 ) + ) /; ( z "\[Rule]" z 0 ) Condition LegendreP ν μ 2 z ν Gamma -1 μ -1 ν Gamma 1 -1 μ ν -1 1 1 -1 Subscript z 0 -1 -1 1 2 μ Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 -1 1 2 μ Subscript z 0 -1 z 2 -1 1 1 Subscript z 0 1 -1 1 2 μ z -1 Subscript z 0 2 -1 Subscript z 0 1 1 2 μ z -1 Subscript z 0 2 -1 1 2 -1 μ z -1 Subscript z 0 2 -1 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 Gamma -1 ν Gamma ν 1 Hypergeometric2F1Regularized -1 ν ν 1 μ 1 Subscript z 0 1 2 -1 -1 1 Subscript z 0 1 -1 -1 μ z -1 Subscript z 0 2 -1 Subscript z 0 1 -1 μ z -1 Subscript z 0 2 -1 MeijerG ν 1 -1 ν 0 -1 μ 1 2 Subscript z 0 1 1 2 Subscript z 0 2 -1 -1 -1 μ z -1 Subscript z 0 2 -1 1 Subscript z 0 1 -1 -1 μ z -1 Subscript z 0 2 -1 Subscript z 0 1 -1 μ z -1 Subscript z 0 2 -1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 1 Subscript z 0 1 -1 μ z -1 Subscript z 0 2 -1 Subscript z 0 1 μ z -1 Subscript z 0 2 -1 Gamma 1 -1 ν Gamma ν 1 Subscript z 0 2 -1 Hypergeometric2F1Regularized 1 -1 ν ν 2 μ 2 Subscript z 0 1 2 -1 -1 2 μ Gamma -1 ν Gamma ν 2 Hypergeometric2F1Regularized -1 ν ν 1 μ 1 Subscript z 0 1 2 -1 2 μ z -1 Subscript z 0 2 -1 μ MeijerG ν 1 -1 ν 0 -1 μ Subscript z 0 1 2 -1 Subscript z 0 2 -1 μ z -1 Subscript z 0 2 -1 MeijerG ν -1 ν -1 0 -1 μ -1 Subscript z 0 1 2 -1 z -1 Subscript z 0 Rule z Subscript z 0 [/itex]

 Rule Form

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

 2007-05-02