html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 WhittakerW

 http://functions.wolfram.com/07.45.20.0002.01

 Input Form

 Derivative[1, 0, 0][WhittakerW][\[Nu], \[Mu], z] == (1/((-1 + 4 \[Mu]^2) Gamma[1/2 - \[Mu] - \[Nu]] Gamma[1/2 + \[Mu] - \[Nu]])) ((z^(3/2 - \[Mu]) (1 + 2 \[Mu]) Gamma[2 \[Mu]] Gamma[1/2 - \[Mu] - \[Nu]] HypergeometricPFQ[{{3/2 - \[Mu] - \[Nu]}, {1}, {1, 1/2 - \[Mu] - \[Nu]}}, {{2, 2 - 2 \[Mu]}, {}, {3/2 - \[Mu] - \[Nu]}}, z, z])/E^(z/2) - (z^(3/2 + \[Mu]) (-1 + 2 \[Mu]) Gamma[-2 \[Mu]] Gamma[1/2 + \[Mu] - \[Nu]] HypergeometricPFQ[{{3/2 + \[Mu] - \[Nu]}, {1}, {1, 1/2 + \[Mu] - \[Nu]}}, {{2, 2 + 2 \[Mu]}, {}, {3/2 + \[Mu] - \[Nu]}}, z, z])/E^(z/2) + (4 \[Mu]^2 - 1) Gamma[1/2 + \[Mu] - \[Nu]] (Gamma[-2 \[Mu]] (PolyGamma[1/2 - \[Mu] - \[Nu]] - PolyGamma[1/2 + \[Mu] - \[Nu]]) WhittakerM[\[Nu], \[Mu], z] + Gamma[1/2 - \[Mu] - \[Nu]] PolyGamma[1/2 + \[Mu] - \[Nu]] WhittakerW[\[Nu], \[Mu], z])) /; !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[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["3", "/", "2"]], "-", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", RowBox[List["2", " ", "\[Mu]"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "}"]]]], "}"]], ",", "z", ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Mu]"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "}"]]]], "}"]], ",", "z", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", SuperscriptBox["\[Mu]", "2"]]], "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", 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["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", 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["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 ) 1 ( 4 μ 2 - 1 ) Γ ( 1 2 - μ - ν ) Γ ( μ - ν + 1 2 ) ( - z 2 ( 2 μ + 1 ) Γ ( 2 μ ) Γ ( - μ - ν + 1 2 ) z 3 2 - μ F 2 0 1 1 1 2 ( 3 2 - μ - ν ; 1 ; 1 , 1 2 - μ - ν ; 2 , 2 - 2 μ ; ; 3 2 - μ - ν ; z , z ) - - z 2 ( 2 μ - 1 ) Γ ( - 2 μ ) Γ ( μ - ν + 1 2 ) z μ + 3 2 F 2 0 1 1 1 2 ( 3 2 + μ - ν ; 1 ; 1 , 1 2 + μ - ν ; 2 , 2 + 2 μ ; ; 3 2 + μ - ν ; z , z ) + ( 4 μ 2 - 1 ) Γ ( μ - ν + 1 2 ) ( Γ ( - 2 μ ) ( ψ TagBox["\[Psi]", PolyGamma] ( - μ - ν + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( μ - ν + 1 2 ) ) M TagBox["M", WhittakerM] ν , μ ( z ) + Γ ( - 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]] FormBox RowBox RowBox RowBox SubsuperscriptBox TagBox W BesselJ RowBox ν , μ TagBox RowBox ( RowBox 1 , 0 , 0 ) Derivative ( z ) RowBox FractionBox 1 RowBox RowBox ( RowBox RowBox 4 SuperscriptBox μ 2 - 1 ) RowBox Γ ( RowBox FractionBox 1 2 - μ - ν ) RowBox Γ ( RowBox μ - ν + FractionBox 1 2 ) RowBox ( RowBox RowBox SuperscriptBox RowBox - FractionBox z 2 RowBox ( RowBox RowBox 2 μ + 1 ) RowBox Γ ( RowBox 2 μ ) RowBox Γ ( RowBox RowBox - μ - ν + FractionBox 1 2 ) SuperscriptBox z RowBox FractionBox 3 2 - μ RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 1 1 2 [ RowBox GridBox ErrorBox RowBox RowBox RowBox FractionBox 3 2 - μ - ν ; 1 ; 1 , RowBox RowBox FractionBox 1 2 - μ - ν ; RowBox 2 , RowBox RowBox RowBox 2 - RowBox 2 μ ; ; RowBox FractionBox 3 2 - μ - ν ; z , z ] - RowBox SuperscriptBox RowBox - FractionBox z 2 RowBox ( RowBox RowBox 2 μ - 1 ) RowBox Γ ( RowBox RowBox - 2 μ ) RowBox Γ ( RowBox μ - ν + FractionBox 1 2 ) SuperscriptBox z RowBox μ + FractionBox 3 2 RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 1 1 2 ( RowBox RowBox GridBox RowBox RowBox RowBox FractionBox 3 2 + μ - ν ; 1 ; 1 , RowBox RowBox FractionBox 1 2 + μ - ν ; RowBox 2 , RowBox RowBox RowBox 2 + RowBox 2 μ ; ; RowBox FractionBox 3 2 + μ - ν ; z , z ) + RowBox RowBox ( RowBox RowBox 4 SuperscriptBox μ 2 - 1 ) RowBox Γ ( RowBox μ - ν + FractionBox 1 2 ) RowBox ( RowBox RowBox RowBox Γ ( RowBox RowBox - 2 μ ) RowBox ( RowBox RowBox TagBox ψ PolyGamma ( RowBox RowBox - μ - ν + FractionBox 1 2 ) - RowBox TagBox ψ PolyGamma ( RowBox μ - ν + FractionBox 1 2 ) ) RowBox SubscriptBox TagBox M WhittakerM RowBox ν , μ ( z ) + RowBox RowBox Γ ( RowBox RowBox - μ - ν + FractionBox 1 2 ) RowBox TagBox ψ PolyGamma ( RowBox μ - ν + FractionBox 1 2 ) RowBox SubscriptBox TagBox W WhittakerW RowBox ν , μ ( z ) ) ) /; RowBox RowBox 2 μ TagBox Function TraditionalForm [/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[FractionBox[RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", RowBox[List["2", " ", "\[Mu]"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "}"]]]], "}"]], ",", "z", ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Mu]"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "}"]]]], "}"]], ",", "z", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]], "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", 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["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", 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[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", 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