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WhittakerW




Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerW[nu,mu,z] > Series representations > Generalized power series > Expansions at z==0 > Logarithmic case





http://functions.wolfram.com/07.45.06.0015.01









  


  








Input Form





WhittakerW[\[Nu], 0, z] == (-(Sqrt[z]/Gamma[1/2 - \[Nu]])) (Log[z] Sum[(((-1)^(-j + k) 2^(j - k) Pochhammer[1/2 - \[Nu], j])/ (j!^2 (-j + k)!)) z^k, {k, 0, Infinity}, {j, 0, k}] - Sum[(((-1)^(-j + k) 2^(j - k) Pochhammer[1/2 - \[Nu], j])/ (j!^2 (-j + k)!)) (2 PolyGamma[1 + j] - PolyGamma[1/2 + j - \[Nu]]) z^k, {k, 0, Infinity}, {j, 0, k}])








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, WhittakerW] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mi> z </mi> </msqrt> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <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> <mo> &#8290; </mo> <mtext> </mtext> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> ! 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "0", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List["j", "-", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "j"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["j", "!"]], ")"]], "2"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]]]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List["j", "-", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j"]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "j", "-", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["j", "!"]], ")"]], "2"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], "]"]]]]]]]]]


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





2007-05-02