Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











WhittakerW




Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerW[nu,mu,z] > Differentiation > Symbolic differentiation > With respect to z





http://functions.wolfram.com/07.45.20.0010.01









  


  








Input Form





D[WhittakerW[\[Nu], \[Mu], z], {z, n}] == Sum[((-1)^(Subscript[k, 1] + Subscript[k, 3]) KroneckerDelta[n, Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3]] Multinomial[Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]] z^(-Subscript[k, 2] - Subscript[k, 3]/2) Pochhammer[3/2 - Subscript[k, 2] + \[Mu], Subscript[k, 2]] Pochhammer[1/2 + \[Mu] - \[Nu], Subscript[k, 3]] WhittakerW[-(Subscript[k, 3]/2) + \[Nu], Subscript[k, 3]/2 + \[Mu], z])/ 2^Subscript[k, 1], {Subscript[k, 1], 0, n}, {Subscript[k, 2], 0, n}, {Subscript[k, 3], 0, n}] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "3"]]]], RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], SuperscriptBox["2", RowBox[List["-", SubscriptBox["k", "1"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", SubscriptBox["k", "2"]]], "-", FractionBox[SubscriptBox["k", "3"], "2"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", SubscriptBox["k", "2"], "+", "\[Mu]"]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", SubscriptBox["k", "3"]]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["k", "3"], "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox[SubscriptBox["k", "3"], "2"], "+", "\[Mu]"]], ",", "z"]], "]"]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mrow> <msub> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, WhittakerW] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </mrow> </mrow> </msub> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;k&quot;, &quot;3&quot;]]], &quot;;&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;,&quot;, SubscriptBox[&quot;k&quot;, &quot;3&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mfrac> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Mu]&quot;, &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;2&quot;]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;3&quot;]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, WhittakerW] </annotation> </semantics> <mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mfrac> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mn> 2 </mn> </mfrac> </mrow> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> WhittakerW </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <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> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> WhittakerW </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "3"]]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", SubscriptBox["k", "1"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", SubscriptBox["k", "2"]]], "-", FractionBox[SubscriptBox["k", "3"], "2"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", SubscriptBox["k", "2"], "+", "\[Mu]"]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", SubscriptBox["k", "3"]]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["k", "3"], "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox[SubscriptBox["k", "3"], "2"], "+", "\[Mu]"]], ",", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02