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 WhittakerW

 http://functions.wolfram.com/07.45.20.0014.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "0", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]"]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], "]"]]]]], RowBox[List["(", " ", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[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[SuperscriptBox[RowBox[List["(", RowBox[List["j", "!"]], " ", ")"]], "2"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]]]]], RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List["k", "+", FractionBox["1", "2"]]], ",", "\[Alpha]"]], "]"]], SuperscriptBox["z", "k"]]]]]]], " ", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[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["Gamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], " "]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["j", "!"]], ")"]], "2"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]], 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"]]]]]]]]], ")"]]]]]]]]]]

 MathML Form

 α W TagBox["W", WhittakerW] ν , 0 ( z ) z α - z 1 2 - α Γ ( 1 2 - ν ) ( k = 0 j = 0 k ( - 1 ) k - j 2 j - k ( 1 2 - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( j ! ) 2 ( k - j ) ! ℱ𝒞 log ( α ) ( z , k + 1 2 ) z k - k = 0 j = 0 k ( - 1 ) k - j 2 j - k ( 1 2 - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "j"], Pochhammer] Γ ( k + 3 2 ) ( j ! ) 2 ( k - j ) ! Γ ( k - α + 3 2 ) ( 2 ψ TagBox["\[Psi]", PolyGamma] ( j + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j - ν + 1 2 ) ) z k ) z α WhittakerW ν 0 z -1 z 1 2 -1 α Gamma 1 2 -1 ν -1 j 0 k k 0 -1 k -1 j 2 j -1 k Pochhammer 1 2 -1 ν j j 2 k -1 j -1 Subscript ℱ𝒞 log α z k 1 2 z k -1 j 0 k k 0 -1 k -1 j 2 j -1 k Pochhammer 1 2 -1 ν j Gamma k 3 2 j 2 k -1 j Gamma k -1 α 3 2 -1 2 PolyGamma j 1 -1 PolyGamma j -1 ν 1 2 z k [/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]_", ",", "0", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[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["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List["k", "+", FractionBox["1", "2"]]], ",", "\[Alpha]"]], "]"]], " ", 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["Gamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]]]], ")"]], " ", 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["k", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], "]"]]]]]]]]]

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

 2007-05-02