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 WhittakerW

 http://functions.wolfram.com/07.45.21.0005.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) WhittakerW[\[Nu], \[Mu], t])/E^(c t), {t, 0, Infinity}] == (-((Pi^2 Csc[2 Pi \[Mu]] Gamma[-\[Alpha] - \[Nu]] (-Tan[Pi (\[Alpha] + \[Mu])] + Tan[Pi (\[Alpha] - \[Mu])]))/ (Gamma[1/2 - \[Alpha] - \[Mu]] Gamma[1/2 - \[Alpha] + \[Mu]] Gamma[1/2 - \[Mu] - \[Nu]] Gamma[1/2 + \[Mu] - \[Nu]]))) Hypergeometric2F1[1/2 + \[Alpha] + \[Mu], 1/2 + \[Alpha] - \[Mu], 1 + \[Alpha] + \[Nu], 1/2 + c] + ((Pi Csc[Pi (\[Alpha] + \[Nu])])/Gamma[1 - \[Alpha] - \[Nu]]) (1/2 + c)^(-\[Alpha] - \[Nu]) Hypergeometric2F1[1/2 + \[Mu] - \[Nu], 1/2 - \[Mu] - \[Nu], 1 - \[Alpha] - \[Nu], 1/2 + c] /; Re[c] > -(1/2) && Re[\[Alpha] + \[Mu]] > -(1/2) && Re[\[Alpha] - \[Mu]] > -(1/2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "t"]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]"]], ")"]]]], "]"]]]], "+", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Mu]"]], ")"]]]], "]"]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]", "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]", "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]], ")"]]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Alpha]", "+", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Alpha]", "-", "\[Mu]"]], ",", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "c"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]", "-", "\[Nu]"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Alpha]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "c"]]]], "]"]]]]]]]], " ", "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "c", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "-", "\[Mu]"]], "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]]]]]]]]

 MathML Form

 0 t α - 1 - c t W TagBox["W", WhittakerW] ν , μ ( t ) t π csc ( π ( α + ν ) ) Γ ( - α - ν + 1 ) ( c + 1 2 ) - α - ν 2 F 1 ( μ - ν + 1 2 , - μ - ν + 1 2 ; - α - ν + 1 ; c + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["c", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] - π 2 csc ( 2 π μ ) Γ ( - α - ν ) ( tan ( π ( α - μ ) ) - tan ( π ( α + μ ) ) ) Γ ( - α - μ + 1 2 ) Γ ( - α + μ + 1 2 ) Γ ( - μ - ν + 1 2 ) Γ ( μ - ν + 1 2 ) 2 F 1 ( α + μ + 1 2 , α - μ + 1 2 ; α + ν + 1 ; c + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Alpha]", "+", "\[Mu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", "\[Mu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Alpha]", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["c", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] /; Re ( c ) > - 1 2 Re ( α + μ ) > - 1 2 Re ( α - μ ) > - 1 2 Condition t 0 t α -1 -1 c t WhittakerW ν μ t α ν Gamma -1 α -1 ν 1 -1 c 1 2 -1 α -1 ν Hypergeometric2F1 μ -1 ν 1 2 -1 μ -1 ν 1 2 -1 α -1 ν 1 c 1 2 -1 2 2 μ Gamma -1 α -1 ν α -1 μ -1 α μ Gamma -1 α -1 μ 1 2 Gamma -1 α μ 1 2 Gamma -1 μ -1 ν 1 2 Gamma μ -1 ν 1 2 -1 Hypergeometric2F1 α μ 1 2 α -1 μ 1 2 α ν 1 c 1 2 c -1 1 2 α μ -1 1 2 α -1 μ -1 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c_"]], " ", "t_"]]], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]"]], ")"]]]], "]"]]]], "+", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Mu]"]], ")"]]]], "]"]]]], ")"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Alpha]", "+", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Alpha]", "-", "\[Mu]"]], ",", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "c"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]", "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]", "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Nu]"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Alpha]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "c"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]", "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "c", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "-", "\[Mu]"]], "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]]]]]]]]]]

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

 2007-05-02