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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Differentiation > Low-order differentiation > With respect to mu





http://functions.wolfram.com/07.11.20.0007.01









  


  










Input Form





D[LegendreQ[\[Nu], \[Mu], 2, z], \[Mu]] == ((Pi Csc[\[Mu] Pi] Gamma[1 + \[Mu] + \[Nu]])/(4 Gamma[1 - \[Mu] + \[Nu]])) (Log[1 + z] - Log[1 - z] + 2 (Pi Cot[\[Mu] Pi] + PolyGamma[1 + \[Mu]] - PolyGamma[1 - \[Mu] + \[Nu]] - PolyGamma[1 + \[Mu] + \[Nu]])) LegendreP[\[Nu], -\[Mu], 2, z] + (Pi/4) (Cot[\[Mu] Pi] (Log[1 + z] - Log[1 - z] + 2 PolyGamma[1 - \[Mu]]) - 2 Pi Csc[\[Mu] Pi]^2) LegendreP[\[Nu], \[Mu], 2, z] + ((Pi \[Nu] (1 + \[Nu]) (z - 1))/4) ((Cot[\[Mu] Pi]/((1 - \[Mu]) Gamma[2 - \[Mu]])) ((1 + z)^(\[Mu]/2)/ (1 - z)^(\[Mu]/2)) HypergeometricPFQ[{{1 - \[Nu], 2 + \[Nu]}, {1}, {1, 1 - \[Mu]}}, {{2, 2 - \[Mu]}, {}, {2 - \[Mu]}}, (1 - z)/2, (1 - z)/2] + ((Csc[\[Mu] Pi] Gamma[1 + \[Mu] + \[Nu]])/ ((1 + \[Mu]) Gamma[2 + \[Mu]] Gamma[1 - \[Mu] + \[Nu]])) ((1 - z)^(\[Mu]/2)/(1 + z)^(\[Mu]/2)) HypergeometricPFQ[ {{1 - \[Nu], 2 + \[Nu]}, {1}, {1, 1 + \[Mu]}}, {{2, 2 + \[Mu]}, {}, {2 + \[Mu]}}, (1 - z)/2, (1 - z)/2]) /; !Element[\[Mu], Integers]










Standard Form





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MathML Form







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</mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#956; </mi> <mo> &#8713; </mo> <mi> &#8484; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> Q </ms> <ci> LegendreQ </ci> </apply> <ms> &#957; </ms> <ms> &#956; </ms> </apply> <ms> ( </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <apply> <ci> HoldComplete </ci> <ci> LegendreQ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <ms> &#956; </ms> </list> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> &#957; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> cot </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> / </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> / </ms> <ms> 2 </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 0 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 1 </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#957; </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> <ms> 1 </ms> <ms> ; </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> csc </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> + </ms> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> / </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> / </ms> <ms> 2 </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 0 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 1 </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#957; </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> <ms> 1 </ms> <ms> ; </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> + </ms> <ms> &#956; </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> cot </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> P </ms> <ci> LegendreP </ci> </apply> <ms> &#957; </ms> <ms> &#956; </ms> </apply> <ms> ( </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> csc </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> - </ms> <ms> &#956; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> cot </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> - </ms> <ms> &#956; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> P </ms> <ci> LegendreP </ci> </apply> <ms> &#957; </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#956; </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#8713; </ms> <ms> &#8484; </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.