Whittaker hypergeometric function W: Integration (formula 07.45.21.0005)

WhittakerW

$Mathematica Notation$

Hypergeometric Functions

WhittakerW[nu,mu,z]

Integration

Definite integration

Involving the direct function

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

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MathML Form

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</apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> α </ci> <ci> μ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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