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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Differentiation > Low-order differentiation > With respect to mu





http://functions.wolfram.com/07.12.20.0007.01









  


  










Input Form





D[LegendreQ[\[Nu], \[Mu], 3, z], \[Mu]] == ((E^(I \[Mu] Pi) Pi Csc[\[Mu] Pi])/2) Pochhammer[1 - \[Mu] + \[Nu], 2 \[Mu]] (Pi Cot[\[Mu] Pi] - I Pi + (1/2) (Log[z + 1] - Log[z - 1]) + PolyGamma[1 + \[Mu]] - PolyGamma[1 - \[Mu] + \[Nu]] - PolyGamma[1 + \[Mu] + \[Nu]]) LegendreP[\[Nu], -\[Mu], 3, z] - ((E^(I \[Mu] Pi) Pi Csc[\[Mu] Pi])/2) (Pi Cot[\[Mu] Pi] - I Pi - (1/2) (Log[z + 1] - Log[z - 1]) - PolyGamma[1 - \[Mu]]) LegendreP[\[Nu], \[Mu], 3, z] + ((E^(I \[Mu] Pi) Pi \[Nu] (1 + \[Nu]) Csc[\[Mu] Pi])/4) ((z - 1)^(\[Mu]/2 + 1)/(z + 1)^(\[Mu]/2)) ((1/((1 - \[Mu]) Gamma[2 - \[Mu]])) ((z + 1)^\[Mu]/(z - 1)^\[Mu]) HypergeometricPFQ[{{1 - \[Nu], 2 + \[Nu]}, {1}, {1, 1 - \[Mu]}}, {{2, 2 - \[Mu]}, {}, {2 - \[Mu]}}, (1 - z)/2, (1 - z)/2] + (Gamma[1 + \[Mu] + \[Nu]]/((1 + \[Mu]) Gamma[2 + \[Mu]] Gamma[1 - \[Mu] + \[Nu]])) 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> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; 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</mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;\[Mu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Mu]&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <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> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </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> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </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> <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> &#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; 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Rule Form





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





2001-10-29





© 1998-2014 Wolfram Research, Inc.