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 WhittakerW

 http://functions.wolfram.com/07.45.06.0005.01

 Input Form

 WhittakerW[\[Nu], \[Mu], z] == Sum[(1/k!) Sum[((-1)^(Subscript[k, 1] + Subscript[k, 3]) KroneckerDelta[k, Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3]] Multinomial[Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]] Pochhammer[3/2 - Subscript[k, 2] + \[Mu], Subscript[k, 2]] Pochhammer[1/2 + \[Mu] - \[Nu], Subscript[k, 3]] x^(-Subscript[k, 2] - Subscript[k, 3]/2) (WhittakerW[-(Subscript[k, 3]/2) + \[Nu], Subscript[k, 3]/2 + \[Mu], x]/E^((2 \[Mu] + 1) Pi I Floor[Arg[z - x]/(2 Pi)]) - E^((2 \[Mu] + 1) Pi I Floor[Arg[z - x]/(2 Pi)]) ((2 I Pi (-1)^Subscript[k, 3] E^(2 \[Mu] Pi I))/ (Gamma[1/2 - \[Mu] - \[Nu]] Gamma[1 + Subscript[k, 3] + 2 \[Mu]])) Floor[Arg[z - x]/(2 Pi)] WhittakerM[-(Subscript[k, 3]/2) + \[Nu], Subscript[k, 3]/2 + \[Mu], x]) (z - x)^k)/2^Subscript[k, 1], {Subscript[k, 1], 0, k}, {Subscript[k, 2], 0, k}, {Subscript[k, 3], 0, k}], {k, 0, Infinity}] /; Element[x, Reals] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "3"]]]], RowBox[List["KroneckerDelta", "[", RowBox[List["k", ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], SuperscriptBox["2", RowBox[List["-", SubscriptBox["k", "1"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", SubscriptBox["k", "2"], "+", "\[Mu]"]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", SubscriptBox["k", "2"]]], "-", FractionBox[SubscriptBox["k", "3"], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]]]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["k", "3"], "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox[SubscriptBox["k", "3"], "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], FractionBox[RowBox[List["2", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["k", "3"]], SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["k", "3"], "+", RowBox[List["2", " ", "\[Mu]"]]]], "]"]]]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["k", "3"], "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox[SubscriptBox["k", "3"], "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 W TagBox["W", WhittakerW] ν , μ ( z ) k = 0 1 k ! k 1 = 0 k k 2 = 0 k k 3 = 0 k ( - 1 ) k 1 + k 3 δ KroneckerDelta k , k 1 + k 2 + k 3 ( k 1 + k 2 + k 3 ; k 1 , k 2 , k 3 ) TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]], ";", SubscriptBox["k", "1"]]], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], ")"]], Multinomial, Rule[Editable, False]] 2 - k 1 ( μ - k 2 + 3 2 ) k 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", SubscriptBox["k", "2"], "+", FractionBox["3", "2"]]], ")"]], SubscriptBox["k", "2"]], Pochhammer] ( μ - ν + 1 2 ) k 3 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], SubscriptBox["k", "3"]], Pochhammer] x - k 2 - k 3 2 ( - ( 2 μ + 1 ) π arg ( z - x ) 2 π W TagBox["W", WhittakerW] ν - k 3 2 , μ + k 3 2 ( x ) - 2 π ( - 1 ) k 3 2 μ π Γ ( 1 2 - μ - ν ) Γ ( 2 μ + k 3 + 1 ) ( 2 μ + 1 ) π arg ( z - x ) 2 π arg ( z - x ) 2 π M TagBox["M", WhittakerM] ν - k 3 2 , μ + k 3 2 ( x ) ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition WhittakerW ν μ z k 0 1 k -1 Subscript k 3 0 k Subscript k 2 0 k Subscript k 1 0 k -1 Subscript k 1 Subscript k 3 KroneckerDelta k Subscript k 1 Subscript k 2 Subscript k 3 Multinomial Subscript k 1 Subscript k 2 Subscript k 3 2 -1 Subscript k 1 Pochhammer μ -1 Subscript k 2 3 2 Subscript k 2 Pochhammer μ -1 ν 1 2 Subscript k 3 x -1 Subscript k 2 -1 Subscript k 3 2 -1 -1 2 μ 1 z -1 x 2 -1 WhittakerW ν -1 Subscript k 3 2 -1 μ Subscript k 3 2 -1 x -1 2 -1 Subscript k 3 2 μ Gamma 1 2 -1 μ -1 ν Gamma 2 μ Subscript k 3 1 -1 2 μ 1 z -1 x 2 -1 z -1 x 2 -1 WhittakerM ν -1 Subscript k 3 2 -1 μ Subscript k 3 2 -1 x z -1 x k x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "3"]]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["k", ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", SubscriptBox["k", "1"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", SubscriptBox["k", "2"], "+", "\[Mu]"]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", SubscriptBox["k", "2"]]], "-", FractionBox[SubscriptBox["k", "3"], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]]]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["k", "3"], "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox[SubscriptBox["k", "3"], "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["k", "3"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["k", "3"], "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox[SubscriptBox["k", "3"], "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["k", "3"], "+", RowBox[List["2", " ", "\[Mu]"]]]], "]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2007-05-02