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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/07.12.06.0022.01









  


  










Input Form





LegendreQ[\[Nu], m, 3, z] \[Proportional] ((-1)^m/(2^(m/2) (4 m! Gamma[-m - \[Nu]] Gamma[1 - m + \[Nu]]))) (Pi^2 + 2 Pi^2 Cot[Pi \[Nu]]^2 - Log[-1 - z] - (1 + 4 Pi Cot[Pi \[Nu]] - 4 Log[1 + z]) Log[1 + z] - 2 (Pi Cos[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) (EulerGamma - PolyGamma[1 + m]) - 2 (Pi Cot[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) (PolyGamma[m] + PolyGamma[m - \[Nu]])) (z + 1)^(m/2) (1 + O[z + 1]) - ((((-1)^m 2^(m/2 - 1) (m - 1)! Sin[Pi \[Nu]])/Pi) (Pi Cot[Pi \[Nu]] + Log[-1 - z] - Log[1 + z]) (1 + O[z + 1]))/ (z + 1)^(m/2) /; (z -> -1) && Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <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> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; 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</mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <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> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </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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</ci> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </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> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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