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 WhittakerW

 http://functions.wolfram.com/07.45.20.0001.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0", ",", "0"]], "]"]], "[", "WhittakerW", "]"]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Mu]", "-", "\[Nu]"]], "]"]], SuperscriptBox["z", "k"]]]]]]], "-", " ", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "\[Mu]", "-", "\[Nu]"]], "]"]], SuperscriptBox["z", "k"]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "\[Mu]"]], ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["WhittakerW", TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], 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