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 WhittakerW

 http://functions.wolfram.com/07.45.20.0013.01

 Input Form

 D[WhittakerW[\[Nu], \[Mu], z], {z, \[Alpha]}] == (Gamma[-2 \[Mu]]/Gamma[1/2 - \[Mu] - \[Nu]]) Sum[Sum[((-1)^(k - j) 2^(-k + j) Pochhammer[\[Mu] - \[Nu] + 1/2, j] Gamma[k + \[Mu] + 3/2])/((k - j)! Pochhammer[2 \[Mu] + 1, j] j! Gamma[k + \[Mu] - \[Alpha] + 3/2]), {j, 0, k}] z^(k + \[Mu] + 1/2 - \[Alpha]), {k, 0, Infinity}] + (Gamma[2 \[Mu]]/Gamma[1/2 + \[Mu] - \[Nu]]) Sum[Sum[((-1)^(k - j) 2^(-k + j) Pochhammer[1/2 - \[Mu] - \[Nu], j] Gamma[k - \[Mu] + 3/2])/((k - j)! Pochhammer[1 - 2 \[Mu], j] j! Gamma[k - \[Mu] - \[Alpha] + 3/2]), {j, 0, k}] z^(k - \[Mu] + 1/2 - \[Alpha]), {k, 0, Infinity}] /; !Element[2 \[Mu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "k"]], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ",", "j"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "j"]], "]"]], RowBox[List["j", "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]]]], ")"]], SuperscriptBox["z", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "k"]], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "j"]], "]"]], RowBox[List["j", "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]]]], ")"]], SuperscriptBox["z", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]]]]]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "\[Mu]"]], ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 α W TagBox["W", WhittakerW] ν , μ ( z ) z α Γ ( 2 μ ) Γ ( μ - ν + 1 2 ) k = 0 ( j = 0 k ( - 1 ) k - j 2 j - k ( - μ - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] Γ ( k - μ + 3 2 ) ( k - j ) ! ( 1 - 2 μ ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "j"], Pochhammer] j ! Γ ( k - α - μ + 3 2 ) ) z k - α - μ + 1 2 + Γ ( - 2 μ ) Γ ( - μ - ν + 1 2 ) k = 0 ( j = 0 k ( - 1 ) k - j 2 j - k ( μ - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] Γ ( k + μ + 3 2 ) ( k - j ) ! ( 2 μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], "j"], Pochhammer] j ! Γ ( k - α + μ + 3 2 ) ) z k - α + μ + 1 2 /; 2 μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z α WhittakerW ν μ z Gamma 2 μ Gamma μ -1 ν 1 2 -1 k 0 j 0 k -1 k -1 j 2 j -1 k Pochhammer -1 μ -1 ν 1 2 j Gamma k -1 μ 3 2 k -1 j Pochhammer 1 -1 2 μ j j Gamma k -1 α -1 μ 3 2 -1 z k -1 α -1 μ 1 2 Gamma -2 μ Gamma -1 μ -1 ν 1 2 -1 k 0 j 0 k -1 k -1 j 2 j -1 k Pochhammer μ -1 ν 1 2 j Gamma k μ 3 2 k -1 j Pochhammer 2 μ 1 j j Gamma k -1 α μ 3 2 -1 z k -1 α μ 1 2 2 μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "k"]], "+", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "k"]], "+", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02