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/07.45.20.0004.01

 Input Form

 Derivative[0, 1, 0][WhittakerW][\[Nu], \[Mu], z] == (-((z^(3/2 - \[Mu]) Gamma[2 \[Mu]])/(E^(z/2) ((1 - 2 \[Mu])^2 Gamma[1/2 + \[Mu] - \[Nu]])))) ((-1 + 2 \[Mu] + 2 \[Nu]) HypergeometricPFQ[{{3/2 - \[Mu] - \[Nu]}, {1}, {1, 1 - 2 \[Mu]}}, {{2, 2 - 2 \[Mu]}, {}, {2 - 2 \[Mu]}}, z, z] + (1 - 2 \[Mu]) HypergeometricPFQ[{{3/2 - \[Mu] - \[Nu]}, {1}, {1, 1/2 - \[Mu] - \[Nu]}}, {{2, 2 - 2 \[Mu]}, {}, {3/2 - \[Mu] - \[Nu]}}, z, z]) - ((z^(3/2 + \[Mu]) Gamma[-2 \[Mu]])/(E^(z/2) ((1 + 2 \[Mu])^2 Gamma[1/2 - \[Mu] - \[Nu]]))) ((1 + 2 \[Mu] - 2 \[Nu]) HypergeometricPFQ[{{3/2 + \[Mu] - \[Nu]}, {1}, {1, 1 + 2 \[Mu]}}, {{2, 2 + 2 \[Mu]}, {}, {2 + 2 \[Mu]}}, z, z] - (1 + 2 \[Mu]) HypergeometricPFQ[{{3/2 + \[Mu] - \[Nu]}, {1}, {1, 1/2 + \[Mu] - \[Nu]}}, {{2, 2 + 2 \[Mu]}, {}, {3/2 + \[Mu] - \[Nu]}}, z, z]) + (1/Gamma[1/2 - \[Mu] - \[Nu]]) (Gamma[-2 \[Mu]] (2 Log[z] - 2 PolyGamma[1 - 2 \[Mu]] - 2 PolyGamma[1 + 2 \[Mu]] + PolyGamma[1/2 - \[Mu] - \[Nu]] + PolyGamma[1/2 + \[Mu] - \[Nu]]) WhittakerM[\[Nu], \[Mu], z] - Gamma[1/2 - \[Mu] - \[Nu]] (2 Pi Cot[2 Pi \[Mu]] + Log[z] - 2 PolyGamma[1 - 2 \[Mu]] + 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["0", ",", "1", ",", "0"]], "]"]], "[", "WhittakerW", "]"]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " "]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", 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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["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Mu]"]], "]"]]]], "+", RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], "]"]]]], "+", 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 ( 0 , 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]], Derivative] ( z ) - - z 2 z 3 2 - μ Γ ( 2 μ ) ( 1 - 2 μ ) 2 Γ ( μ - ν + 1 2 ) ( ( 2 μ + 2 ν - 1 ) F 2 0 1 1 1 2 ( 3 2 - μ - ν ; 1 ; 1 , 1 - 2 μ ; 2 , 2 - 2 μ ; ; 2 - 2 μ ; z , z ) + ( 1 - 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 z μ + 3 2 Γ ( - 2 μ ) ( 2 μ + 1 ) 2 Γ ( 1 2 - μ - ν ) ( ( 2 μ - 2 ν + 1 ) F 2 0 1 1 1 2 ( 3 2 + μ - ν ; 1 ; 1 , 1 + 2 μ ; 2 , 2 + 2 μ ; ; 2 + 2 μ ; z , z ) - ( 2 μ + 1 ) 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 ) ) + 1 Γ ( 1 2 - μ - ν ) ( Γ ( - 2 μ ) ( 2 log ( z ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( 2 μ + 1 ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( 1 - 2 μ ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - μ - ν ) + ψ TagBox["\[Psi]", PolyGamma] ( μ - ν + 1 2 ) ) M TagBox["M", WhittakerM] ν , μ ( z ) - Γ ( 1 2 - μ - ν ) ( 2 π cot ( 2 π μ ) + log ( z ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( 1 - 2 μ ) + 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 0 , 1 , 0 ) Derivative ( z ) RowBox RowBox RowBox - FractionBox RowBox SuperscriptBox RowBox - FractionBox z 2 SuperscriptBox z RowBox FractionBox 3 2 - μ RowBox Γ ( RowBox 2 μ ) RowBox SuperscriptBox RowBox ( RowBox 1 - RowBox 2 μ ) 2 RowBox Γ ( RowBox μ - ν + FractionBox 1 2 ) RowBox ( RowBox RowBox RowBox ( RowBox RowBox 2 μ + RowBox 2 ν - 1 ) 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 1 - RowBox 2 μ ; RowBox 2 , RowBox RowBox RowBox 2 - RowBox 2 μ ; ; RowBox 2 - RowBox 2 μ ; z , z ] + RowBox RowBox ( RowBox 1 - RowBox 2 μ ) 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 FractionBox RowBox SuperscriptBox RowBox - FractionBox z 2 SuperscriptBox z RowBox μ + FractionBox 3 2 RowBox Γ ( RowBox RowBox - 2 μ ) RowBox SuperscriptBox RowBox ( RowBox RowBox 2 μ + 1 ) 2 RowBox Γ ( RowBox FractionBox 1 2 - μ - ν ) RowBox ( RowBox RowBox RowBox ( RowBox RowBox 2 μ - RowBox 2 ν + 1 ) 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 1 + RowBox 2 μ ; RowBox 2 , RowBox RowBox RowBox 2 + RowBox 2 μ ; ; RowBox 2 + RowBox 2 μ ; z , z ) - RowBox RowBox ( RowBox RowBox 2 μ + 1 ) 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 FractionBox 1 RowBox Γ ( RowBox FractionBox 1 2 - μ - ν ) RowBox ( RowBox RowBox RowBox Γ ( RowBox RowBox - 2 μ ) RowBox ( RowBox RowBox 2 RowBox log ( z ) - RowBox 2 RowBox TagBox ψ PolyGamma ( RowBox RowBox 2 μ + 1 ) - RowBox 2 RowBox TagBox ψ PolyGamma ( RowBox 1 - RowBox 2 μ ) + RowBox TagBox ψ PolyGamma ( RowBox FractionBox 1 2 - μ - ν ) + RowBox TagBox ψ PolyGamma ( RowBox μ - ν + FractionBox 1 2 ) ) RowBox SubscriptBox TagBox M WhittakerM RowBox ν , μ ( z ) - RowBox RowBox Γ ( RowBox FractionBox 1 2 - μ - ν ) RowBox ( RowBox RowBox 2 π RowBox cot ( RowBox 2 π μ ) + RowBox log ( z ) - RowBox 2 RowBox TagBox ψ PolyGamma ( RowBox 1 - RowBox 2 μ ) + 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["0", ",", "1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "}"]], ",", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02