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WhittakerW






Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerW[nu,mu,z] > Specific values > Specialized values > For fixed z and integer parameters





http://functions.wolfram.com/07.45.03.0031.01









  


  










Input Form





WhittakerW[-n - m/2, -((m + 1)/2), z] == ((m + 1) E^(z/2) Sum[(((-1)^q z^(n + m - p - q) (m + k)!)/ (p! k! (n - p - q - 1)! (m + n - p)! (q - k)!)) (Sum[z^j/Pochhammer[q - k - m - 1, m - q + k + 2 + j], {j, 0, q - k - m - 2}]/E^z - Sum[z^j/Pochhammer[q - k - m - 1, m - q + k + 2 + j], {j, q - k - m - 1, -1}]/E^z + ((-1)^(m + q - k)/(m + 1 - q + k)!) (ExpIntegralEi[-z] - (1/2) (Log[-z] - Log[-(1/z)]) + Log[z])), {p, 0, n - 1}, {q, 0, n - 1}, {k, 0, q}])/z^(m/2) /; Element[n, Integers] && n > 0 && Element[m, Integers] && m >= 0










Standard Form





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MathML Form







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type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> j </ci> <ci> k </ci> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "-", FractionBox["m_", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["m_", "+", "1"]], ")"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox["m", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "q"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox["z", RowBox[List["n", "+", "m", "-", "p", "-", "q"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "k"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["q", "-", "k", "-", "m", "-", "2"]]], FractionBox[SuperscriptBox["z", "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["q", "-", "k", "-", "m", "-", "1"]], ",", RowBox[List["m", "-", "q", "+", "k", "+", "2", "+", "j"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["q", "-", "k", "-", "m", "-", "1"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox["z", "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["q", "-", "k", "-", "m", "-", "1"]], ",", RowBox[List["m", "-", "q", "+", "k", "+", "2", "+", "j"]]]], "]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "q", "-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["-", FractionBox["1", "z"]]], "]"]]]], ")"]]]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1", "-", "q", "+", "k"]], ")"]], "!"]]]]], ")"]]]], RowBox[List[RowBox[List["p", "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "p", "-", "q", "-", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "-", "p"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["q", "-", "k"]], ")"]], "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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