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 LegendreP

 http://functions.wolfram.com/07.07.06.0029.01

 Input Form

 LegendreP[\[Nu], z] \[Proportional] (Tan[Pi \[Nu]]/2) ((Gamma[-\[Nu]]/(2^\[Nu] Gamma[1 + \[Nu]])) (2 I Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] (-1 - Subscript[z, 0])^ \[Nu] + Csc[Pi \[Nu]] (1 + Subscript[z, 0])^\[Nu]) Hypergeometric2F1Regularized[-\[Nu], -\[Nu], -2 \[Nu], 2/(1 + Subscript[z, 0])] - ((2^(1 + \[Nu]) Gamma[1 + \[Nu]])/ Gamma[-\[Nu]]) (2 I Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] (-1 - Subscript[z, 0])^ (-1 - \[Nu]) + Csc[Pi \[Nu]] (1 + Subscript[z, 0])^(-1 - \[Nu])) Hypergeometric2F1Regularized[1 + \[Nu], 1 + \[Nu], 2 + 2 \[Nu], 2/(1 + Subscript[z, 0])] - (1/(1 + Subscript[z, 0])) ((-((2^(1 + \[Nu]) Gamma[2 + \[Nu]])/Gamma[-\[Nu]])) (2 I Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] (-1 - Subscript[z, 0])^ (-1 - \[Nu]) + Csc[Pi \[Nu]] (1 + Subscript[z, 0])^(-1 - \[Nu])) Hypergeometric2F1Regularized[1 + \[Nu], 2 + \[Nu], 2 + 2 \[Nu], 2/(1 + Subscript[z, 0])] + (Gamma[1 - \[Nu]]/ (2^\[Nu] Gamma[1 + \[Nu]])) (2 I Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] (-1 - Subscript[z, 0])^ \[Nu] + Csc[Pi \[Nu]] (1 + Subscript[z, 0])^\[Nu]) Hypergeometric2F1Regularized[1 - \[Nu], -\[Nu], -2 \[Nu], 2/(1 + Subscript[z, 0])]) (z - Subscript[z, 0]) + \[Ellipsis]) /; (z -> Subscript[z, 0])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", 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["2", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["z", 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FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], "+", RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]]]], ")"]], 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", "+", SubscriptBox["z", "0"]]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["1", "+", SubscriptBox["z", "0"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", 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", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["z", "0"]]], ")"]], "\[Nu]"]]], "+", RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], "\[Nu]"]]]]], ")"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", FractionBox["2", 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 ( z ) 1 2 tan ( π ν ) ( 2 - ν Γ ( - ν ) Γ ( ν + 1 ) 2 F ~ 1 ( - ν , - ν ; - 2 ν ; 2 z 0 + 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[SubscriptBox["z", "0"], "+", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ( 2 arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π ( - z 0 - 1 ) ν + csc ( π ν ) ( z 0 + 1 ) ν ) + - 2 ν + 1 Γ ( ν + 1 ) Γ ( - ν ) ( 2 arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π ( - z 0 - 1 ) - ν - 1 + csc ( π ν ) ( z 0 + 1 ) - ν - 1 ) 2 F ~ 1 ( ν + 1 , ν + 1 ; 2 ν + 2 ; 2 z 0 + 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[SubscriptBox["z", "0"], "+", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - 1 z 0 + 1 ( 2 - ν Γ ( 1 - ν ) Γ ( ν + 1 ) ( 2 arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π ( - z 0 - 1 ) ν + csc ( π ν ) ( z 0 + 1 ) ν ) 2 F ~ 1 ( 1 - ν , - ν ; - 2 ν ; 2 z 0 + 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["1", "-", "\[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[SubscriptBox["z", "0"], "+", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - 2 ν + 1 Γ ( ν + 2 ) Γ ( - ν ) ( 2 arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π ( - z 0 - 1 ) - ν - 1 + csc ( π ν ) ( z 0 + 1 ) - ν - 1 ) 2 F ~ 1 ( ν + 1 , ν + 2 ; 2 ν + 2 ; 2 z 0 + 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]", "+", "2"]], 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[SubscriptBox["z", "0"], "+", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) ( z - z 0 ) + ) /; ( z "\[Rule]" z 0 ) Condition Proportional LegendreP ν z 1 2 ν 2 -1 ν Gamma -1 ν Gamma ν 1 -1 Hypergeometric2F1Regularized -1 ν -1 ν -2 ν 2 Subscript z 0 1 -1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 -1 Subscript z 0 -1 ν ν Subscript z 0 1 ν -1 2 ν 1 Gamma ν 1 Gamma -1 ν -1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 -1 Subscript z 0 -1 -1 ν -1 ν Subscript z 0 1 -1 ν -1 Hypergeometric2F1Regularized ν 1 ν 1 2 ν 2 2 Subscript z 0 1 -1 -1 1 Subscript z 0 1 -1 2 -1 ν Gamma 1 -1 ν Gamma ν 1 -1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 -1 Subscript z 0 -1 ν ν Subscript z 0 1 ν Hypergeometric2F1Regularized 1 -1 ν -1 ν -2 ν 2 Subscript z 0 1 -1 -1 2 ν 1 Gamma ν 2 Gamma -1 ν -1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 -1 Subscript z 0 -1 -1 ν -1 ν Subscript z 0 1 -1 ν -1 Hypergeometric2F1Regularized ν 1 ν 2 2 ν 2 2 Subscript z 0 1 -1 z -1 Subscript z 0 Rule z Subscript z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", 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["2", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02