Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
 











WhittakerM






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.44.20.0009.01









  


  










Input Form





Derivative[n, 0, 0][WhittakerM][\[Nu], \[Mu], z] == z^(\[Mu] + 1/2) Sum[Sum[(((-1)^(k - j) 2^(-k + j))/(k - j)!) (1/(Pochhammer[2 \[Mu] + 1, j] j!)) D[Pochhammer[\[Mu] - \[Nu] + 1/2, j], {\[Nu], n}], {j, 0, k}] z^k, {k, 0, Infinity}] /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["n", ",", "0", ",", "0"]], "]"]], "[", "WhittakerM", "]"]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "k"]], "+", "j"]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]], FractionBox["1", RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "j"]], "]"]], RowBox[List["j", "!"]]]]]]], RowBox[List["D", "[", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ",", "j"]], "]"]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]]], "]"]]]]]], ")"]], SuperscriptBox["z", "k"]]]]]]]]], "/;", 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> <mrow> <msubsup> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox[&quot;M&quot;, BesselJ] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> <semantics> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;,&quot;, &quot;0&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Mu]&quot;]], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <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> <mi> j </mi> </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;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#957; </mi> <mi> n </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <ci> n </ci> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </list> <apply> <ci> Subscript </ci> <apply> <ci> BesselJ </ci> <ci> M </ci> </apply> <ci> &#957; </ci> <ci> &#956; </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <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> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <ci> n </ci> </degree> </bvar> <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> j </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["WhittakerM", TagBox[RowBox[List["(", RowBox[List["n", ",", "0", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "k"]], "+", "j"]]]]], ")"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ",", "j"]], "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]]]], ")"]]]]]]], ")"]], " ", SuperscriptBox["z", "k"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.