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WhittakerW






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.45.20.0002.01









  


  










Input Form





Derivative[1, 0, 0][WhittakerW][\[Nu], \[Mu], z] == (1/((-1 + 4 \[Mu]^2) Gamma[1/2 - \[Mu] - \[Nu]] Gamma[1/2 + \[Mu] - \[Nu]])) ((z^(3/2 - \[Mu]) (1 + 2 \[Mu]) Gamma[2 \[Mu]] Gamma[1/2 - \[Mu] - \[Nu]] HypergeometricPFQ[{{3/2 - \[Mu] - \[Nu]}, {1}, {1, 1/2 - \[Mu] - \[Nu]}}, {{2, 2 - 2 \[Mu]}, {}, {3/2 - \[Mu] - \[Nu]}}, z, z])/E^(z/2) - (z^(3/2 + \[Mu]) (-1 + 2 \[Mu]) Gamma[-2 \[Mu]] Gamma[1/2 + \[Mu] - \[Nu]] HypergeometricPFQ[{{3/2 + \[Mu] - \[Nu]}, {1}, {1, 1/2 + \[Mu] - \[Nu]}}, {{2, 2 + 2 \[Mu]}, {}, {3/2 + \[Mu] - \[Nu]}}, z, z])/E^(z/2) + (4 \[Mu]^2 - 1) Gamma[1/2 + \[Mu] - \[Nu]] (Gamma[-2 \[Mu]] (PolyGamma[1/2 - \[Mu] - \[Nu]] - PolyGamma[1/2 + \[Mu] - \[Nu]]) WhittakerM[\[Nu], \[Mu], z] + Gamma[1/2 - \[Mu] - \[Nu]] PolyGamma[1/2 + \[Mu] - \[Nu]] WhittakerW[\[Nu], \[Mu], z])) /; !Element[2 \[Mu], Integers]










Standard Form





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MathML Form







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</ms> <ms> - </ms> <ms> &#957; </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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> &#956; </ms> <ms> - </ms> <ms> &#957; </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> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#956; </ms> </list> </apply> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> &#956; </ms> <ms> - </ms> <ms> &#957; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <ms> z </ms> </list> </apply> <ms> , </ms> <ms> z </ms> <ms> ] </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#8519; </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#956; </ms> </list> </apply> <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> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <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> &#956; </ms> <ms> - </ms> <ms> &#957; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> </list> </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> 1 </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> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> &#956; </ms> <ms> - </ms> <ms> &#957; </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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> &#956; </ms> <ms> - </ms> <ms> &#957; </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> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#956; </ms> </list> </apply> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> &#956; </ms> <ms> - </ms> <ms> &#957; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <ms> z </ms> </list> </apply> <ms> , </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#956; </ms> <ms> 2 </ms> </apply> </list> </apply> <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> &#957; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> &#956; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> M </ms> <ci> WhittakerM </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> , </ms> <ms> &#956; </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> W </ms> <ci> WhittakerW </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> , </ms> <ms> &#956; </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#956; </ms> </list> </apply> <ms> &#8713; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </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)





2007-05-02





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