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/07.10.20.0006.01

 Input Form

 D[LegendreQ[\[Nu], z], {z, \[Alpha]}] == (-((Sin[Pi \[Nu]] z^(1 - \[Alpha]))/(2 Pi))) Sum[((1 + (-1)^j)/2^j) Gamma[j - \[Nu]] Gamma[1 + j + \[Nu]] HypergeometricPFQRegularized[{{j - \[Nu], 1 + \[Nu] + j}, {}, {2 + j}}, {{1 + j}, {}, {1, 1 + j, 2 + j - \[Alpha]}}, 1/2, -(z/2)] z^j, {j, 0, Infinity}] - (PolyGamma[\[Nu] + 1] HypergeometricPFQRegularized[ {{-\[Nu], 1 + \[Nu]}, {}, {1}}, {{1}, {}, {1 - \[Alpha]}}, 1/2, -(z/2)])/z^\[Alpha] + Sum[(((-1)^j Pochhammer[-\[Nu], k + j] Pochhammer[\[Nu] + 1, k + j])/ ((k + j)! k! Gamma[j - \[Alpha] + 1] 2^(k + j))) PolyGamma[k + j + 1] z^(j - \[Alpha]), {k, 0, Infinity}, {j, 0, Infinity}] /; Abs[z] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]]]], RowBox[List["2", "\[Pi]", " "]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"]]], SuperscriptBox["2", "j"]], RowBox[List["Gamma", "[", RowBox[List["j", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "j", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["j", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]", "+", "j"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "j"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "j"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "j"]], ",", RowBox[List["2", "+", "j", "-", "\[Alpha]"]]]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]], SuperscriptBox["z", "j"]]]]]]], " ", "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "+", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["k", "+", "j"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]], "!"]], RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List["j", "-", "\[Alpha]", "+", "1"]], "]"]], SuperscriptBox["2", RowBox[List["k", "+", "j"]]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "j", "+", "1"]], "]"]], SuperscriptBox["z", RowBox[List["j", "-", "\[Alpha]"]]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]

 MathML Form

 α Q TagBox["Q", LegendreQ] ν ( z ) z α - ψ TagBox["\[Psi]", PolyGamma] ( ν + 1 ) z - α F ~ 1 0 1 2 0 1 ( - ν , ν + 1 ; ; 1 ; 1 ; ; 1 - α ; 1 2 , - z 2 ) - z 1 - α sin ( π ν ) 2 π j = 0 1 + ( - 1 ) j 2 j Γ ( j - ν ) Γ ( j + ν + 1 ) F ~ 1 0 3 2 0 1 ( j - ν , j + ν + 1 ; ; j + 2 ; j + 1 ; ; 1 , j + 1 , j - α + 2 ; 1 2 , - z 2 ) z j + k = 0 j = 0 ( - 1 ) j ( - ν ) j + k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] ( ν + 1 ) j + k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 ) z j - α ( j + k ) ! k ! Γ ( j - α + 1 ) 2 j + k /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 FormBox RowBox RowBox FractionBox RowBox SuperscriptBox α RowBox SubscriptBox TagBox Q LegendreQ ν ( z ) RowBox SuperscriptBox z α RowBox RowBox RowBox - RowBox TagBox ψ PolyGamma ( RowBox ν + 1 ) SuperscriptBox z RowBox - α RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 0 1 RowBox 2 0 1 [ RowBox GridBox ErrorBox RowBox RowBox - ν , RowBox RowBox RowBox ν + 1 ; ; 1 ; RowBox RowBox 1 ; ; RowBox 1 - α ; FractionBox 1 2 , RowBox - FractionBox z 2 ] - RowBox FractionBox RowBox SuperscriptBox z RowBox 1 - α RowBox sin ( RowBox π ν ) RowBox 2 π RowBox UnderoverscriptBox RowBox j = 0 RowBox FractionBox RowBox 1 + SuperscriptBox RowBox ( RowBox - 1 ) j SuperscriptBox 2 j RowBox Γ ( RowBox j - ν ) RowBox Γ ( RowBox j + ν + 1 ) RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 0 3 RowBox 2 0 1 ( RowBox RowBox GridBox RowBox RowBox j - ν , RowBox RowBox RowBox j + ν + 1 ; ; RowBox j + 2 ; RowBox RowBox RowBox RowBox j + 1 ; ; 1 , RowBox j + 1 , RowBox RowBox j - α + 2 ; FractionBox 1 2 , RowBox - FractionBox z 2 ) SuperscriptBox z j + RowBox UnderoverscriptBox RowBox k = 0 RowBox UnderoverscriptBox RowBox j = 0 FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) j TagBox SubscriptBox RowBox ( RowBox - ν ) RowBox j + k Pochhammer TagBox SubscriptBox RowBox ( RowBox ν + 1 ) RowBox j + k Pochhammer RowBox TagBox ψ PolyGamma ( RowBox j + k + 1 ) SuperscriptBox z RowBox j - α RowBox RowBox RowBox ( RowBox j + k ) ! RowBox k ! RowBox Γ ( RowBox j - α + 1 ) SuperscriptBox 2 RowBox j + k /; RowBox RowBox z < 1 TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["j", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "j", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["j", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]", "+", "j"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "j"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "j"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "j"]], ",", RowBox[List["2", "+", "j", "-", "\[Alpha]"]]]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]], " ", SuperscriptBox["z", "j"]]], SuperscriptBox["2", "j"]]]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "+", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["k", "+", "j"]]]], "]"]]]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "j", "+", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["j", "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]], "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["j", "-", "\[Alpha]", "+", "1"]], "]"]], " ", SuperscriptBox["2", RowBox[List["k", "+", "j"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]

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

 2001-10-29