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

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], ")"]]]], "-", "EulerGamma", "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </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> <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> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <ci> &#957; </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29