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WhittakerW




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.45.20.0003.01









  


  








Input Form





Derivative[0, 1, 0][WhittakerW][\[Nu], \[Mu], z] == ((2 Gamma[-2 \[Mu]])/Gamma[1/2 - \[Mu] - \[Nu]]) Log[z] WhittakerM[\[Nu], \[Mu], z] - (2 Pi Cot[2 Pi \[Mu]] + Log[z] - PolyGamma[1/2 - \[Mu] - \[Nu]] + PolyGamma[1/2 + \[Mu] - \[Nu]]) WhittakerW[\[Nu], \[Mu], z] + ((Gamma[-2 \[Mu]]/Gamma[1/2 - \[Mu] - \[Nu]]) z^(1/2 + \[Mu]) Sum[(Pochhammer[1/2 + \[Mu] - \[Nu], k]/(Pochhammer[1 + 2 \[Mu], k] k!)) (-2 PolyGamma[1 + k + 2 \[Mu]] + PolyGamma[1/2 + k + \[Mu] - \[Nu]]) z^k, {k, 0, Infinity}])/E^(z/2) - ((Gamma[2 \[Mu]]/Gamma[1/2 + \[Mu] - \[Nu]]) z^(1/2 - \[Mu]) Sum[(Pochhammer[1/2 - \[Mu] - \[Nu], k]/(Pochhammer[1 - 2 \[Mu], k] k!)) (-2 PolyGamma[1 + k - 2 \[Mu]] + PolyGamma[1/2 + k - \[Mu] - \[Nu]]) z^k, {k, 0, Infinity}])/E^(z/2) /; !Element[2 \[Mu], Integers]








Standard Form





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MathML Form







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</ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> log </ci> <ci> z </ci> <apply> <ci> WhittakerM </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <cot /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> WhittakerW </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02