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 LegendreQ

 http://functions.wolfram.com/07.11.06.0036.01

 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]) Sum[((Pochhammer[-\[Nu], k] Pochhammer[m - \[Nu], k])/ (Pochhammer[-2 \[Nu], k] k!)) (2/(1 - z))^k, {k, 0, Infinity}] + Gamma[-(1/2) - \[Nu]] Gamma[1 - m + \[Nu]] (2 Pi Cot[Pi \[Nu]] + Log[1 + z] - Log[-z - 1]) Sum[((Pochhammer[1 + \[Nu], k] Pochhammer[1 + m + \[Nu], k])/ (Pochhammer[2 + 2 \[Nu], k] k!)) (2/(1 - z))^k, {k, 0, Infinity}]) /; Abs[(1 - z)/2] > 1 && 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", "\[Nu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], 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]] ) 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 k = 0 ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( m - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( 2 1 - z ) k + Γ ( - ν - 1 2 ) Γ ( ν - m + 1 ) ( 2 π cot ( π ν ) + log ( 1 + z ) - log ( - z - 1 ) ) k = 0 ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( m + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( 2 ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], "k"], Pochhammer] k ! ( 2 1 - z ) k ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" > 1 m Condition LegendreQ ν m 2 z 2 -1 ν -2 1 2 Gamma -1 m -1 ν Gamma ν -1 m 1 -1 z -1 -1 ν -1 1 z m 2 -1 1 -1 z m 2 -1 -1 2 2 ν 1 Gamma -1 m -1 ν Gamma ν 1 2 1 z -1 -1 z -1 z -1 2 ν 1 k 0 Pochhammer -1 ν k Pochhammer m -1 ν k Pochhammer -2 ν k k -1 2 1 -1 z -1 k Gamma -1 ν -1 1 2 Gamma ν -1 m 1 2 ν 1 z -1 -1 z -1 k 0 Pochhammer ν 1 k Pochhammer m ν 1 k Pochhammer 2 ν 2 k k -1 2 1 -1 z -1 k 1 -1 z 2 -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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], ")"]]]], 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["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], ">", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29