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/07.11.06.0002.01

 Input Form

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

 Standard Form

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 MathML Form

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", " ", "z"]], ")"]], " ", 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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[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["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", 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"], " ", RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["\[Mu]", " ", "z"]], ")"]]]], ")"]], " ", 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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[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["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", 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