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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Series representations > Generalized power series > Expansions at z==1





http://functions.wolfram.com/07.12.06.0011.01









  


  










Input Form





LegendreQ[\[Nu], m, 3, z] \[Proportional] ((-1)^m 2^(m/2 - 1) (m - 1)! (1 + O[z - 1]))/(z - 1)^(m/2) + ((2^(-2 - m/2) Gamma[1 + m + \[Nu]])/(I^m (m m! Gamma[1 - m + \[Nu]]))) (1 - z)^(m/2) (2 + m (Log[2/(z - 1)] - 2 PolyGamma[1 - m + \[Nu]] + 2 EulerGamma - 2 PolyGamma[1 + m + \[Nu]])) (1 + O[z - 1]) /; (z -> 1) && Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> &#120084; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "m_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["2", RowBox[List[FractionBox["m", "2"], "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", FractionBox["m", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "/", "2"]]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["2", RowBox[List["z", "-", "1"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", "EulerGamma"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]], RowBox[List["m", " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m", "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]










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





2001-10-29





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