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 WhittakerW

 http://functions.wolfram.com/07.45.20.0012.01

 Input Form

 D[WhittakerW[\[Nu], \[Mu], z], {z, \[Alpha]}] == (Gamma[-2 \[Mu]]/Gamma[1/2 - \[Mu] - \[Nu]]) Sum[(((-(1/2))^k Gamma[k + \[Mu] + 3/2])/ (k! Gamma[k + \[Mu] - \[Alpha] + 3/2])) Hypergeometric2F1[-k, 1/2 + \[Mu] - \[Nu], 1 + 2 \[Mu], 2] z^(k + \[Mu] + 1/2 - \[Alpha]), {k, 0, Infinity}] + (Gamma[2 \[Mu]]/Gamma[1/2 + \[Mu] - \[Nu]]) Sum[(((-(1/2))^k Gamma[k - \[Mu] + 3/2])/ (k! Gamma[k - \[Mu] - \[Alpha] + 3/2])) Hypergeometric2F1[-k, 1/2 - \[Mu] - \[Nu], 1 + 2 \[Mu], 2] 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[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "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[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "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 ( - 1 2 ) k Γ ( k - μ + 3 2 ) k ! Γ ( k - α - μ + 3 2 ) 2 F 1 ( - k , - μ - ν + 1 2 ; 2 μ + 1 ; 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["2", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] z k - α - μ + 1 2 + Γ ( - 2 μ ) Γ ( 1 2 - μ - ν ) k = 0 ( - 1 2 ) k Γ ( k + μ + 3 2 ) k ! Γ ( k - α + μ + 3 2 ) 2 F 1 ( - k , μ - ν + 1 2 ; 2 μ + 1 ; 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["2", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] 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 -1 1 2 k Gamma k -1 μ 3 2 k Gamma k -1 α -1 μ 3 2 -1 Hypergeometric2F1 -1 k -1 μ -1 ν 1 2 2 μ 1 2 z k -1 α -1 μ 1 2 Gamma -2 μ Gamma 1 2 -1 μ -1 ν -1 k 0 -1 1 2 k Gamma k μ 3 2 k Gamma k -1 α μ 3 2 -1 Hypergeometric2F1 -1 k μ -1 ν 1 2 2 μ 1 2 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]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "2"]], "]"]], " ", SuperscriptBox["z", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]]]]]], 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]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "2"]], "]"]], " ", SuperscriptBox["z", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "-", "\[Alpha]", "+", FractionBox["3", "2"]]], "]"]]]]]]]]], 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