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 WhittakerM

 http://functions.wolfram.com/07.44.06.0006.01

 Input Form

 WhittakerM[\[Nu], \[Mu], z] \[Proportional] Exp[(2 \[Mu] + 1) Pi I Floor[Arg[z - x]/(2 Pi)]] (WhittakerM[\[Nu], \[Mu], x] + (1/2) (((1 + 2 \[Mu] - 2 \[Nu]) WhittakerM[-(1/2) + \[Nu], 1/2 + \[Mu], x])/((1 + 2 \[Mu]) Sqrt[x]) + ((1 + 2 \[Mu] - x) WhittakerM[\[Nu], \[Mu], x])/x) (z - x) + (1/8) ((2 (1 + 2 \[Mu] - 2 \[Nu]) WhittakerM[-(1/2) + \[Nu], 1/2 + \[Mu], x])/x^(3/2) - (2 (1 + 2 \[Mu] - 2 \[Nu]) WhittakerM[-(1/2) + \[Nu], 1/2 + \[Mu], x])/((1 + 2 \[Mu]) Sqrt[x]) + WhittakerM[\[Nu], \[Mu], x] + ((-1 + 4 \[Mu]^2) WhittakerM[\[Nu], \[Mu], x])/x^2 + (1/(2 x)) ((((1 + 2 \[Mu] - 2 \[Nu]) (3 + 2 \[Mu] - 2 \[Nu]))/ (1 + 3 \[Mu] + 2 \[Mu]^2)) WhittakerM[-1 + \[Nu], 1 + \[Mu], x] - 4 (1 + 2 \[Mu]) WhittakerM[\[Nu], \[Mu], x])) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Exp", "[", 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[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", SqrtBox["x"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", "x"]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]]]], "x"]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], SuperscriptBox["x", RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", SqrtBox["x"]]]], "+", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]]]], SuperscriptBox["x", "2"]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "x"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " "]], RowBox[List["1", "+", RowBox[List["3", " ", "\[Mu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]]], RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", "x"]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 M TagBox["M", WhittakerM] ν , μ ( z ) ( 2 μ + 1 ) π arg ( z - x ) 2 π ( M TagBox["M", WhittakerM] ν , μ ( x ) + 1 2 ( - x + 2 μ + 1 x M TagBox["M", WhittakerM] ν , μ ( x ) + 2 μ - 2 ν + 1 ( 2 μ + 1 ) x M TagBox["M", WhittakerM] ν - 1 2 , μ + 1 2 ( x ) ) ( z - x ) + 1 8 ( 4 μ 2 - 1 x 2 M TagBox["M", WhittakerM] ν , μ ( x ) + M TagBox["M", WhittakerM] ν , μ ( x ) + 1 2 x ( ( 2 μ - 2 ν + 1 ) ( 2 μ - 2 ν + 3 ) 2 μ 2 + 3 μ + 1 M TagBox["M", WhittakerM] ν - 1 , μ + 1 ( x ) - 4 ( 2 μ + 1 ) M TagBox["M", WhittakerM] ν , μ ( x ) ) + 2 ( 2 μ - 2 ν + 1 ) x 3 / 2 M TagBox["M", WhittakerM] ν - 1 2 , μ + 1 2 ( x ) - 2 ( 2 μ - 2 ν + 1 ) ( 2 μ + 1 ) x M TagBox["M", WhittakerM] ν - 1 2 , μ + 1 2 ( x ) ) ( z - x ) 2 + ) /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition Proportional WhittakerM ν μ z 2 μ 1 z -1 x 2 -1 WhittakerM ν μ x 1 2 -1 x 2 μ 1 x -1 WhittakerM ν μ x 2 μ -1 2 ν 1 2 μ 1 x 1 2 -1 WhittakerM ν -1 1 2 μ 1 2 x z -1 x 1 8 4 μ 2 -1 x 2 -1 WhittakerM ν μ x WhittakerM ν μ x 1 2 x -1 2 μ -1 2 ν 1 2 μ -1 2 ν 3 2 μ 2 3 μ 1 -1 WhittakerM ν -1 μ 1 x -1 4 2 μ 1 WhittakerM ν μ x 2 2 μ -1 2 ν 1 x 3 2 -1 WhittakerM ν -1 1 2 μ 1 2 x -1 2 2 μ -1 2 ν 1 2 μ 1 x 1 2 -1 WhittakerM ν -1 1 2 μ 1 2 x z -1 x 2 Rule z x x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[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[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", SqrtBox["x"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", "x"]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]]]], "x"]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], SuperscriptBox["x", RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", SqrtBox["x"]]]], "+", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]]]], SuperscriptBox["x", "2"]], "+", FractionBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", "x"]], "]"]]]], RowBox[List["1", "+", RowBox[List["3", " ", "\[Mu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "x"]], "]"]]]]]], RowBox[List["2", " ", "x"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2007-05-02