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 LegendreP

 http://functions.wolfram.com/07.07.06.0037.01

 Input Form

 LegendreP[\[Nu], z] \[Proportional] (Tan[Pi \[Nu]]/2) ((Gamma[-\[Nu]]/(2^\[Nu] Gamma[1 + \[Nu]])) ((1 + x)^\[Nu] Csc[Pi \[Nu]] + 2 I (-1 - x)^\[Nu] Floor[Arg[-x + z]/(2 Pi)]) Hypergeometric2F1Regularized[-\[Nu], -\[Nu], -2 \[Nu], 2/(1 + x)] - ((2^(1 + \[Nu]) Gamma[1 + \[Nu]])/ Gamma[-\[Nu]]) ((1 + x)^(-1 - \[Nu]) Csc[Pi \[Nu]] + 2 I (-1 - x)^(-1 - \[Nu]) Floor[Arg[-x + z]/(2 Pi)]) Hypergeometric2F1Regularized[1 + \[Nu], 1 + \[Nu], 2 + 2 \[Nu], 2/(1 + x)]) + O[z - x] /; Element[x, Reals] && x < -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "\[Nu]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "x"]], ")"]], "\[Nu]"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", FractionBox["2", RowBox[List["1", "+", "x"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "\[Nu]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "x"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "+", "x"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 P TagBox["P", LegendreP] ν ( z ) 1 2 tan ( π ν ) ( 2 - ν Γ ( - ν ) Γ ( ν + 1 ) ( 2 arg ( z - x ) 2 π ( - x - 1 ) ν + ( x + 1 ) ν csc ( π ν ) ) 2 F ~ 1 ( - ν , - ν ; - 2 ν ; 2 x + 1 ) 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]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["x", "+", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - 2 ν + 1 Γ ( ν + 1 ) Γ ( - ν ) ( 2 arg ( z - x ) 2 π ( - x - 1 ) - ν - 1 + ( x + 1 ) - ν - 1 csc ( π ν ) ) 2 F ~ 1 ( ν + 1 , ν + 1 ; 2 ν + 2 ; 2 x + 1 ) 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]", "+", "1"]], 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[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["x", "+", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) + O ( z - x ) /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < - 1 Condition Proportional LegendreP ν z 1 2 ν 2 -1 ν Gamma -1 ν Gamma ν 1 -1 2 z -1 x 2 -1 -1 x -1 ν x 1 ν ν Hypergeometric2F1Regularized -1 ν -1 ν -2 ν 2 x 1 -1 -1 2 ν 1 Gamma ν 1 Gamma -1 ν -1 2 z -1 x 2 -1 -1 x -1 -1 ν -1 x 1 -1 ν -1 ν Hypergeometric2F1Regularized ν 1 ν 1 2 ν 2 2 x 1 -1 O z -1 x x x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "\[Nu]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "x"]], ")"]], "\[Nu]"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", FractionBox["2", RowBox[List["1", "+", "x"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "x"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "+", "x"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], ")"]]]], "+", RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2007-05-02