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 LegendreQ

 http://functions.wolfram.com/07.11.06.0037.01

 Input Form

 LegendreQ[\[Nu], m, 2, z] == (2^(-\[Nu] - 2)/(Sqrt[Pi] Gamma[-m - \[Nu]] Gamma[1 - m + \[Nu]])) (z - 1)^(-\[Nu] - 1) ((1 + z)^(m/2)/(1 - z)^(m/2)) (2^(2 \[Nu] + 1) Gamma[-m - \[Nu]] Gamma[1/2 + \[Nu]] (z - 1)^(2 \[Nu] + 1) (Log[1 + z] - Log[-z - 1]) Hypergeometric2F1[-\[Nu], m - \[Nu], -2 \[Nu], 2/(1 - z)] + Gamma[-(1/2) - \[Nu]] Gamma[1 - m + \[Nu]] (2 Pi Cot[Pi \[Nu]] + Log[1 + z] - Log[-z - 1]) Hypergeometric2F1[1 + \[Nu], 1 + m + \[Nu], 2 + 2 \[Nu], 2/(1 - z)]) /; !IntervalMemberQ[Interval[{-1, 1}], z] && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "m", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], " "]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["m", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], "]"]]]], ")"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["m", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", " ", "z"]], "]"]], "]"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] ν m ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) 2 - ν - 2 π Γ ( - m - ν ) Γ ( ν - m + 1 ) ( z - 1 ) - ν - 1 ( 1 + z ) m / 2 ( 1 - z ) m / 2 ( 2 2 ν + 1 Γ ( - m - ν ) Γ ( ν + 1 2 ) ( log ( 1 + z ) - log ( - z - 1 ) ) ( z - 1 ) 2 ν + 1 2 F 1 ( - ν , m - ν ; - 2 ν ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + Γ ( - ν - 1 2 ) Γ ( ν - m + 1 ) ( 2 π cot ( π ν ) + log ( 1 + z ) - log ( - z - 1 ) ) 2 F 1 ( ν + 1 , m + ν + 1 ; 2 ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) /; ​​ z ( - 1 , 1 ) m Q TagBox["Q", LegendreQ] ν m ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) 2 - ν - 2 π Γ ( - m - ν ) Γ ( ν - m + 1 ) ( z - 1 ) - ν - 1 ( 1 + z ) m / 2 ( 1 - z ) m / 2 ( 2 2 ν + 1 Γ ( - m - ν ) Γ ( ν + 1 2 ) ( log ( 1 + z ) - log ( - z - 1 ) ) ( z - 1 ) 2 ν + 1 2 F 1 ( - ν , m - ν ; - 2 ν ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + Γ ( - ν - 1 2 ) Γ ( ν - m + 1 ) ( 2 π cot ( π ν ) + log ( 1 + z ) - log ( - z - 1 ) ) 2 F 1 ( ν + 1 , m + ν + 1 ; 2 ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) /; ​​ z ( - 1 , 1 ) m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "m_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["m", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["m", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29