html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 LegendreQ

 http://functions.wolfram.com/07.11.21.0001.01

 Input Form

 Integrate[LegendreQ[\[Nu], \[Mu], 2, z], z] == (Pi/2) Csc[\[Mu] Pi] (Cos[\[Mu] Pi] (\[Mu]/2) z^2 Sum[(k + 1)! Pochhammer[1 + \[Mu]/2, k] z^k HypergeometricPFQRegularized[ {{k + 2}, {-(\[Mu]/2)}, {-\[Nu], 1 + \[Nu], 1 - \[Mu]/2}}, {{k + 3}, {}, {k + 2, 1 - \[Mu]}}, -z, -(z/2)], {k, 0, Infinity}] + z Gamma[1 - \[Mu]/2] Sum[Pochhammer[-(\[Mu]/2), k] (-z)^k HypergeometricPFQRegularized[{{-\[Nu], 1 + \[Nu], 1 - \[Mu]/2}, {1}, {k + 1}}, {{1, 1 - \[Mu]}, {1 - \[Mu]/2}, {k + 2}}, 1/2, -(z/2)], {k, 0, Infinity}] + Pochhammer[1 + \[Nu] - \[Mu], 2 \[Mu]] ((\[Mu]/2) z^2 Sum[(k + 1)! Pochhammer[1 - \[Mu]/2, k] z^k HypergeometricPFQRegularized[{{k + 2}, {\[Mu]/2}, {-\[Nu], 1 + \[Nu], 1 + \[Mu]/2}}, {{k + 3}, {}, {k + 2, 1 + \[Mu]}}, -z, -(z/2)], {k, 0, Infinity}] - z Gamma[1 + \[Mu]/2] Sum[Pochhammer[\[Mu]/2, k] (-z)^k HypergeometricPFQRegularized[{{-\[Nu], 1 + \[Nu], 1 + \[Mu]/2}, {1}, {k + 1}}, {{1, 1 + \[Mu]}, {1 + \[Mu]/2}, {k + 2}}, 1/2, -(z/2)], {k, 0, Infinity}])) /; !Element[\[Mu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", FractionBox[RowBox[List[" ", "\[Mu]"]], "2"], SuperscriptBox["z", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], SuperscriptBox["z", "k"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "3"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List["z", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", "\[Mu]"]], "2"], SuperscriptBox["z", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], SuperscriptBox["z", "k"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "3"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "-", RowBox[List["z", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 Q TagBox["Q", LegendreQ] ν μ ( z TagBox["z", HoldComplete[LegendreQ, 2]] ) z π csc ( μ π ) 2 ( μ cos ( μ π ) 2 z 2 k = 0 ( k + 1 ) ! ( μ 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Mu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z k z k F ~ 1 0 2 1 1 3 ( k + 2 ; - μ 2 ; - ν , ν + 1 , 1 - μ 2 ; k + 3 ; ; k + 2 , 1 - μ ; - z , - z 2 ) + Γ ( 1 - μ 2 ) z k = 0 ( - μ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Mu]", "2"]]], ")"]], "k"], Pochhammer] ( - z ) k F ~ 2 1 1 3 1 1 ( - ν , ν + 1 , 1 - μ 2 ; 1 ; k + 1 ; 1 , 1 - μ ; 1 - μ 2 ; k + 2 ; 1 2 , - z 2 ) + ( ν - μ + 1 ) 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] ( μ 2 z 2 k = 0 ( k + 1 ) ! ( 1 - μ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], ")"]], "k"], Pochhammer] z k F ~ 1 0 2 1 1 3 ( k + 2 ; μ 2 ; - ν , ν + 1 , 1 + μ 2 ; k + 3 ; ; k + 2 , 1 + μ ; - z , - z 2 ) - z Γ ( μ 2 + 1 ) k = 0 ( μ 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["\[Mu]", "2"], ")"]], "k"], Pochhammer] ( - z ) k F ~ 2 1 1 3 1 1 ( - ν , ν + 1 , 1 + μ 2 ; 1 ; k + 1 ; 1 , 1 + μ ; 1 + μ 2 ; k + 2 ; 1 2 , - z 2 ) ) ) /; μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] FormBox RowBox RowBox RowBox RowBox RowBox SubsuperscriptBox TagBox Q LegendreQ ν μ ( TagBox z HoldComplete LegendreQ 2 ) RowBox z RowBox FractionBox RowBox π RowBox csc ( RowBox μ π ) 2 RowBox ( RowBox RowBox FractionBox RowBox μ RowBox cos ( RowBox μ π ) 2 SuperscriptBox z 2 RowBox UnderoverscriptBox RowBox k = 0 ErrorBox RowBox RowBox RowBox ( RowBox k + 1 ) ! TagBox SubscriptBox RowBox ( RowBox FractionBox μ 2 + 1 ) k Pochhammer SuperscriptBox z k SuperscriptBox z k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 0 2 RowBox 1 1 3 ( RowBox RowBox GridBox RowBox RowBox RowBox k + 2 ; RowBox - FractionBox μ 2 ; RowBox - ν , RowBox ν + 1 , RowBox RowBox 1 - FractionBox μ 2 ; RowBox RowBox RowBox RowBox k + 3 ; ; RowBox k + 2 , RowBox RowBox 1 - μ ; - z , RowBox - FractionBox z 2 ) + RowBox RowBox Γ ( RowBox 1 - FractionBox μ 2 ) z RowBox UnderoverscriptBox RowBox k = 0 RowBox TagBox SubscriptBox RowBox ( RowBox - FractionBox μ 2 ) k Pochhammer SuperscriptBox RowBox ( RowBox - z ) k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 2 1 1 RowBox 3 1 1 ( RowBox RowBox GridBox RowBox RowBox - ν , RowBox ν + 1 , RowBox RowBox 1 - FractionBox μ 2 ; 1 ; RowBox k + 1 ; RowBox 1 , RowBox RowBox 1 - μ ; RowBox 1 - FractionBox μ 2 ; RowBox k + 2 ; FractionBox 1 2 , RowBox - FractionBox z 2 ) + RowBox TagBox SubscriptBox RowBox ( RowBox ν - μ + 1 ) RowBox 2 μ Pochhammer RowBox ( RowBox RowBox FractionBox μ 2 SuperscriptBox z 2 RowBox UnderoverscriptBox RowBox k = 0 RowBox RowBox RowBox ( RowBox k + 1 ) ! TagBox SubscriptBox RowBox ( RowBox 1 - FractionBox μ 2 ) k Pochhammer SuperscriptBox z k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 0 2 RowBox 1 1 3 ( RowBox RowBox GridBox RowBox RowBox RowBox k + 2 ; FractionBox μ 2 ; RowBox - ν , RowBox ν + 1 , RowBox RowBox 1 + FractionBox μ 2 ; RowBox RowBox RowBox RowBox k + 3 ; ; RowBox k + 2 , RowBox RowBox 1 + μ ; - z , RowBox - FractionBox z 2 ) - RowBox z RowBox Γ ( RowBox FractionBox μ 2 + 1 ) RowBox UnderoverscriptBox RowBox k = 0 RowBox TagBox SubscriptBox RowBox ( FractionBox μ 2 ) k Pochhammer SuperscriptBox RowBox ( RowBox - z ) k RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 2 1 1 RowBox 3 1 1 ( RowBox RowBox GridBox RowBox RowBox - ν , RowBox ν + 1 , RowBox RowBox 1 + FractionBox μ 2 ; 1 ; RowBox k + 1 ; RowBox 1 , RowBox RowBox 1 + μ ; RowBox 1 + FractionBox μ 2 ; RowBox k + 2 ; FractionBox 1 2 , RowBox - FractionBox z 2 ) ) ) /; RowBox μ TagBox Function TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", "\[Mu]", " ", SuperscriptBox["z", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "3"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List["z", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Mu]", " ", SuperscriptBox["z", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "3"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "-", RowBox[List["z", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["\[Mu]", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]]]], ")"]]]]]], ")"]]]], "/;", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2001-10-29