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variants of this functions
LegendreQ




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Series representations > Generalized power series > Expansions at z==infinity





http://functions.wolfram.com/07.11.06.0040.01









  


  








Input Form





LegendreQ[\[Nu], -m, 2, z] == (-((2^(-\[Nu] - 2) Sin[Pi \[Nu]])/Pi^(3/2))) (z - 1)^(-\[Nu] - 1) ((1 + z)^(m/2)/(1 - z)^(m/2)) (2^(2 \[Nu] + 1) Gamma[-m - \[Nu]] Gamma[1/2 + \[Nu]] (z - 1)^(2 \[Nu] + 1) (Log[1 + z] - Log[-z - 1]) Hypergeometric2F1[-\[Nu], m - \[Nu], -2 \[Nu], 2/(1 - z)] + Gamma[-(1/2) - \[Nu]] Gamma[1 - m + \[Nu]] (2 Pi Cot[Pi \[Nu]] + Log[1 + z] - Log[-z - 1]) Hypergeometric2F1[1 + \[Nu], 1 + m + \[Nu], 2 + 2 \[Nu], 2/(1 - z)]) /; !IntervalMemberQ[Interval[{-1, 1}], z] && Element[m, Integers] && m > 0








Standard Form





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MathML Form







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</mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </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> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; 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Rule Form





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





2001-10-29