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 LegendreP

 http://functions.wolfram.com/07.08.06.0054.01

 Input Form

 LegendreP[\[Nu], \[Mu], 2, z] == ((Sin[Pi \[Nu]]/(Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]])) (x + 1)^((1/2) \[Mu]) Exp[Pi I \[Mu] Floor[Arg[z - x]/(2 Pi)]] Sum[((((-1)^j 2^(i - k))/((i - j)! j! (k - i)!)) Pochhammer[-(\[Mu]/2), j] Pochhammer[\[Mu]/2, i - j] (1 - x)^(j - i) (((2 I Floor[Arg[z - x]/(2 Pi)] Gamma[-i + k - \[Nu]] Gamma[i + \[Nu] + 1])/E^(I Pi \[Mu] Floor[Arg[z - x]/(2 Pi)]) + Csc[Pi \[Mu]] Gamma[-i + k - \[Nu]] Gamma[1 - i + k + \[Nu]] Exp[-2 Pi I \[Mu] Floor[Arg[z - x]/(2 Pi)]]) Hypergeometric2F1Regularized[-i + k - \[Nu], 1 - i + k + \[Nu], 1 - i + k + \[Mu], (1 + x)/2] - 2^(-i + k + \[Mu]) Csc[Pi \[Mu]] (1 + x)^(i - k - \[Mu]) Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]] Exp[-2 Pi I \[Mu] Floor[Arg[z - x]/(2 Pi)]] Hypergeometric2F1Regularized[-\[Mu] - \[Nu], 1 - \[Mu] + \[Nu], 1 + i - k - \[Mu], (1 + x)/2]) (z - x)^k)/(x + 1)^j, {k, 0, Infinity}, {i, 0, k}, {j, 0, i}])/(1 - x)^((1/2) \[Mu]) /; !Element[\[Mu], Integers] && 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 ) Γ ( - μ - ν ) Γ ( - μ + ν + 1 ) ( 1 - x ) - μ 2 ( x + 1 ) μ 2 π μ arg ( z - x ) 2 π k = 0 i = 0 k j = 0 i ( - 1 ) j 2 i - k ( i - j ) ! j ! ( k - i ) ! ( - μ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], "j"], Pochhammer] ( μ 2 ) i - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Mu]", "2"], ")"]], RowBox[List["i", "-", "j"]]], Pochhammer] ( 1 - x ) j - i ( x + 1 ) - j ( ( 2 - π μ arg ( z - x ) 2 π arg ( z - x ) 2 π Γ ( i + ν + 1 ) Γ ( - i + k - ν ) + csc ( π μ ) - 2 π μ arg ( z - x ) 2 π Γ ( - i + k + ν + 1 ) Γ ( - i + k - ν ) ) 2 F ~ 1 ( - i + k - ν , - i + k + ν + 1 ; - i + k + 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[RowBox[List["-", "i"]], "+", "k", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "i"]], "+", "k", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "i"]], "+", "k", "+", "\[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 - i + k + μ csc ( π μ ) ( x + 1 ) i - k - μ Γ ( - μ - ν ) Γ ( - μ + ν + 1 ) - 2 π μ arg ( z - x ) 2 π 2 F ~ 1 ( - μ - ν , - μ + ν + 1 ; i - k - 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[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["i", "-", "k", "-", "\[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] ) ( z - x ) k /; μ TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] 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 j 0 i i 0 k k 0 -1 j 2 i -1 k i -1 j j k -1 i -1 Pochhammer -1 μ 2 -1 j Pochhammer μ 2 -1 i -1 j 1 -1 x j -1 i x 1 -1 j 2 -1 μ z -1 x 2 -1 z -1 x 2 -1 Gamma i ν 1 Gamma -1 i k -1 ν μ -2 μ z -1 x 2 -1 Gamma -1 i k ν 1 Gamma -1 i k -1 ν Hypergeometric2F1Regularized -1 i k -1 ν -1 i k ν 1 -1 i k μ 1 x 1 2 -1 -1 2 -1 i k μ μ x 1 i -1 k -1 μ Gamma -1 μ -1 ν Gamma -1 μ ν 1 -2 μ z -1 x 2 -1 Hypergeometric2F1Regularized -1 μ -1 ν -1 μ ν 1 i -1 k -1 μ 1 x 1 2 -1 z -1 x k μ 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "i"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", RowBox[List["i", "-", "k"]]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["i", "-", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List["j", "-", "i"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", "1"]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Mu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["i", "+", "\[Nu]", "+", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "i", "+", "k", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "i", "+", "k", "+", "\[Nu]"]], ",", RowBox[List["1", "-", "i", "+", "k", "+", "\[Mu]"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "i"]], "+", "k", "+", "\[Mu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List["i", "-", "k", "-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "i", "-", "k", "-", "\[Mu]"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i"]], ")"]], "!"]]]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2007-05-02