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 LegendreQ

 http://functions.wolfram.com/07.11.06.0035.01

 Input Form

 LegendreQ[\[Nu], m, 2, z] == ((2^(\[Nu] - 1) Pi Csc[2 Pi \[Nu]])/(Gamma[-m - \[Nu]] Gamma[-\[Nu]])) (z - 1)^(-1 - \[Nu]) ((1 + z)^(m/2)/(1 - z)^(m/2)) (Log[1 + z] - Log[1 - z] + 2 Pi Cot[Pi \[Nu]]) Hypergeometric2F1Regularized[1 + \[Nu], 1 + m + \[Nu], 2 + 2 \[Nu], 2/(1 - z)] + (((-1)^m 2^(\[Nu] - 1) Pi Csc[2 Pi \[Nu]])/ (Gamma[-m - \[Nu]] Gamma[-\[Nu]])) (z - 1)^(-1 - \[Nu]) ((1 - z)^(m/2)/(1 + z)^(m/2)) (4 Pi Cot[Pi \[Nu]] - 2 PolyGamma[m - \[Nu]] - 2 PolyGamma[1 + m + \[Nu]] + Log[1 + z] - Log[1 - z]) Hypergeometric2F1Regularized[1 + \[Nu], 1 - m + \[Nu], 2 + 2 \[Nu], 2/(1 - z)] - (((-1)^m 2^(-2 - \[Nu]) Pi Csc[2 Pi \[Nu]])/ (Gamma[1 + \[Nu]] Gamma[1 - m + \[Nu]])) (z - 1)^\[Nu] ((1 - z)^(m/2)/(1 + z)^(m/2)) (Log[1 + z] - Log[1 - z] - 2 PolyGamma[-m - \[Nu]] - 2 PolyGamma[1 - m + \[Nu]]) Hypergeometric2F1Regularized[-m - \[Nu], -\[Nu], -2 \[Nu], 2/(1 - z)] - ((2^(-2 - \[Nu]) Pi Csc[2 Pi \[Nu]])/(Gamma[1 + \[Nu]] Gamma[1 - m + \[Nu]])) (z - 1)^\[Nu] ((1 + z)^(m/2)/(1 - z)^(m/2)) (Log[1 + z] - Log[1 - z] - 2 Pi Cot[Pi \[Nu]]) Hypergeometric2F1Regularized[m - \[Nu], -\[Nu], -2 \[Nu], 2/(1 - z)] + ((2^(\[Nu] - 1) Gamma[1/2 + \[Nu]])/(Sqrt[Pi] Gamma[1 - m + \[Nu]])) (z - 1)^\[Nu] ((1 - z)^(m/2)/(1 + z)^(m/2)) Sum[(Pochhammer[-\[Nu], k]/(k! Pochhammer[-2 \[Nu], k])) ((-1)^m Pochhammer[-m - \[Nu], k] PolyGamma[k - m - \[Nu]] + ((1 + z)/(1 - z))^m Pochhammer[m - \[Nu], k] PolyGamma[k + m - \[Nu]]) (2/(1 - z))^k, {k, 0, Infinity}] + ((2^(-\[Nu] - 2) Gamma[-(1/2) - \[Nu]])/(Sqrt[Pi] Gamma[-m - \[Nu]])) (z - 1)^(-\[Nu] - 1) ((1 - z)^(m/2)/(1 + z)^(m/2)) Sum[(Pochhammer[1 + \[Nu], k]/(k! Pochhammer[2 + 2 \[Nu], k])) ((-1)^m Pochhammer[1 - m + \[Nu], k] PolyGamma[1 + k - m + \[Nu]] + ((1 + z)/(1 - z))^m Pochhammer[1 + m + \[Nu], k] PolyGamma[1 + k + m + \[Nu]]) (2/(1 - z))^k, {k, 0, Infinity}] /; Abs[(1 - z)/2] > 1 && Element[m, Integers] && m >= 0

 Standard Form

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" ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "m", "-", "\[Nu]"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], "]"]], " "]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "-", "m", "+", "\[Nu]"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], RowBox[List["1", "-", "z"]]], ")"]], "m"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "m", "+", "\[Nu]"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], ">", "1"]], "\[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]] ) - ( - 1 ) m 2 - ν - 2 π csc ( 2 π ν ) Γ ( ν + 1 ) Γ ( ν - m + 1 ) ( z - 1 ) ν ( 1 - z ) m / 2 ( 1 + z ) m / 2 ( log ( 1 + z ) - log ( 1 - z ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( - m - ν ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( ν - m + 1 ) ) 2 F ~ 1 ( - m - ν , - ν ; - 2 ν ; 2 1 - z ) 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["-", "m"]], "-", "\[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["1", "-", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + ( - 1 ) m 2 ν - 1 π csc ( 2 π ν ) Γ ( - m - ν ) Γ ( - ν ) ( z - 1 ) - ν - 1 ( 1 - z ) m / 2 ( 1 + z ) m / 2 ( log ( 1 + z ) - log ( 1 - z ) + 4 π cot ( π ν ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( m - ν ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( m + ν + 1 ) ) 2 F ~ 1 ( ν + 1 , ν - m + 1 ; 2 ν + 2 ; 2 1 - z ) 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]", "-", "m", "+", "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["1", "-", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - 2 - ν - 2 π csc ( 2 π ν ) Γ ( ν + 1 ) Γ ( ν - m + 1 ) ( z - 1 ) ν ( 1 + z ) m / 2 ( 1 - z ) m / 2 ( log ( 1 + z ) - log ( 1 - z ) - 2 π cot ( π ν ) ) 2 F ~ 1 ( m - ν , - ν ; - 2 ν ; 2 1 - z ) 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["m", "-", "\[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["1", "-", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + 2 ν - 1 π csc ( 2 π ν ) Γ ( - m - ν ) Γ ( - ν ) ( z - 1 ) - ν - 1 ( 1 + z ) m / 2 ( 1 - z ) m / 2 ( 2 π cot ( π ν ) + log ( 1 + z ) - log ( 1 - z ) ) 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[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["m", "+", "\[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["1", "-", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + 2 ν - 1 Γ ( ν + 1 2 ) π Γ ( ν - m + 1 ) ( z - 1 ) ν ( 1 - z ) m / 2 ( 1 + z ) m / 2 k = 0 ( - ν ) k k ! ( - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ( m - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + m - ν ) ( 1 + z 1 - z ) m + ( - 1 ) m ( - m - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k - m - ν ) ) ( 2 1 - z ) k + 2 - ν - 2 Γ ( - ν - 1 2 ) π Γ ( - m - ν ) ( z - 1 ) - ν - 1 ( 1 - z ) m / 2 ( 1 + z ) m / 2 k = 0 ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 2 ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], "k"], Pochhammer] ( ( m + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + m + ν + 1 ) ( 1 + z 1 - z ) m + ( - 1 ) m ( ν - m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "m", "+", "1"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k - m + ν + 1 ) ) ( 2 1 - z ) k /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" > 1 m Condition LegendreQ ν m 2 z -1 -1 m 2 -1 ν -2 2 ν Gamma ν 1 Gamma ν -1 m 1 -1 z -1 ν 1 -1 z m 2 -1 1 z m 2 -1 -1 1 z -1 1 -1 z -1 2 PolyGamma -1 m -1 ν -1 2 PolyGamma ν -1 m 1 Hypergeometric2F1Regularized -1 m -1 ν -1 ν -2 ν 2 1 -1 z -1 -1 m 2 ν -1 2 ν Gamma -1 m -1 ν Gamma -1 ν -1 z -1 -1 ν -1 1 -1 z m 2 -1 1 z m 2 -1 -1 1 z -1 1 -1 z 4 ν -1 2 PolyGamma m -1 ν -1 2 PolyGamma m ν 1 Hypergeometric2F1Regularized ν 1 ν -1 m 1 2 ν 2 2 1 -1 z -1 -1 2 -1 ν -2 2 ν Gamma ν 1 Gamma ν -1 m 1 -1 z -1 ν 1 z m 2 -1 1 -1 z m 2 -1 -1 1 z -1 1 -1 z -1 2 ν Hypergeometric2F1Regularized m -1 ν -1 ν -2 ν 2 1 -1 z -1 2 ν -1 2 ν Gamma -1 m -1 ν Gamma -1 ν -1 z -1 -1 ν -1 1 z m 2 -1 1 -1 z m 2 -1 -1 2 ν 1 z -1 1 -1 z Hypergeometric2F1Regularized ν 1 m ν 1 2 ν 2 2 1 -1 z -1 2 ν -1 Gamma ν 1 2 1 2 Gamma ν -1 m 1 -1 z -1 ν 1 -1 z m 2 -1 1 z m 2 -1 -1 k 0 Subscript -1 ν k k Pochhammer -2 ν k -1 Pochhammer m -1 ν k PolyGamma k m -1 ν 1 z 1 -1 z -1 m -1 m Pochhammer -1 m -1 ν k PolyGamma k -1 m -1 ν 2 1 -1 z -1 k 2 -1 ν -2 Gamma -1 ν -1 1 2 1 2 Gamma -1 m -1 ν -1 z -1 -1 ν -1 1 -1 z m 2 -1 1 z m 2 -1 -1 k 0 Pochhammer ν 1 k k Pochhammer 2 ν 2 k -1 Pochhammer m ν 1 k PolyGamma k m ν 1 1 z 1 -1 z -1 m -1 m Pochhammer ν -1 m 1 k PolyGamma k -1 m ν 1 2 1 -1 z -1 k 1 -1 z 2 -1 1 m [/itex]

 Rule Form

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

 2001-10-29