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 LegendreP

 http://functions.wolfram.com/07.09.06.0040.01

 Input Form

 LegendreP[\[Nu], \[Mu], 3, z] \[Proportional] (((-1)^(2 \[Nu] + 1) 2^(\[Nu] + 1))/((-\[Mu] - \[Nu] - 1)! Gamma[-\[Nu]] Gamma[2 \[Nu] + 2])) (Log[z/2] - PolyGamma[-\[Mu] - \[Nu]] - PolyGamma[1 + \[Nu]] + PolyGamma[2 + 2 \[Nu]] - EulerGamma) z^(-\[Nu] - 1) (1 + O[1/z]) + ((2^\[Nu] Gamma[1/2 + \[Nu]])/ (Sqrt[Pi] Gamma[1 - \[Mu] + \[Nu]])) z^\[Nu] (1 + O[1/z]) + (((-1)^(2 \[Nu]) 2^(1 - \[Mu]) Sin[Pi \[Nu]] Gamma[1 - \[Mu]])/ (Pi Gamma[1 - \[Mu] - \[Nu]] Gamma[2 - \[Mu] + \[Nu]])) z^(\[Mu] - 1) (1 + O[1/z]) /; (Abs[z] -> Infinity) && Element[2 \[Nu] + 1, Integers] && 2 \[Nu] + 1 >= 0 && Element[\[Nu] - \[Mu], Integers] && \[Mu] + \[Nu] <= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "2"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]], "-", "EulerGamma"]], ")"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "\[Mu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox["z", RowBox[List["\[Mu]", "-", "1"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["\[Mu]", "+", "\[Nu]"]], "\[LessEqual]", "0"]]]]]]]]

 MathML Form

 𝔓 ν μ ( z TagBox["z", HoldComplete[LegendreP, 2]] ) ( - 1 ) 2 ν + 1 2 ν + 1 z - ν - 1 ( - μ - ν - 1 ) ! Γ ( - ν ) Γ ( 2 ν + 2 ) ( log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( - μ - ν ) - ψ TagBox["\[Psi]", PolyGamma] ( ν + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 ν + 2 ) - TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ( 1 + O ( 1 z ) ) + 2 ν Γ ( ν + 1 2 ) z ν π Γ ( ν - μ + 1 ) ( 1 + O ( 1 z ) ) + ( - 1 ) 2 ν 2 1 - μ sin ( π ν ) Γ ( 1 - μ ) π Γ ( 1 - μ - ν ) Γ ( 2 - μ + ν ) z μ - 1 ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) 2 ν + 1 ν - μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] μ + ν 0 Condition Proportional Subscript 𝔓 ν μ HoldComplete LegendreP 2 z -1 2 ν 1 2 ν 1 z -1 ν -1 -1 μ -1 ν -1 Gamma -1 ν Gamma 2 ν 2 -1 z 2 -1 -1 PolyGamma -1 μ -1 ν -1 PolyGamma ν 1 PolyGamma 2 ν 2 -1 1 O 1 z -1 2 ν Gamma ν 1 2 z ν 1 2 Gamma ν -1 μ 1 -1 1 O 1 z -1 -1 2 ν 2 1 -1 μ ν Gamma 1 -1 μ Gamma 1 -1 μ -1 ν Gamma 2 -1 μ ν -1 z μ -1 1 O 1 z -1 Rule z 2 ν 1 ν -1 μ μ ν 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]], "-", "EulerGamma"]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "\[Mu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["\[Mu]", "-", "1"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Mu]", "+", "\[Nu]"]], "\[LessEqual]", "0"]]]]]]]]]]

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

 2001-10-29