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 HypergeometricU

 http://functions.wolfram.com/07.33.21.0009.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) HypergeometricU[a, b, t])/E^(t z), {t, 0, Infinity}] == (Pi^2 (Cot[Pi (b - \[Alpha])] + Cot[Pi \[Alpha]]) Csc[b Pi] Gamma[a - \[Alpha]] Hypergeometric2F1[\[Alpha], 1 - b + \[Alpha], 1 - a + \[Alpha], z])/(Gamma[a] Gamma[1 + a - b] Gamma[1 - \[Alpha]] Gamma[b - \[Alpha]]) - (1/Gamma[1 + a - \[Alpha]]) Pi z^(a - \[Alpha]) Csc[Pi (a - \[Alpha])] Hypergeometric2F1[a, 1 + a - b, 1 + a - \[Alpha], z] /; Re[z] > 0 && Re[\[Alpha]] > 0 && Re[1 + \[Alpha] - b] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "t"]], " ", "z"]]], RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", "b", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["b", "-", "\[Alpha]"]], ")"]]]], "]"]], "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Alpha]"]], "]"]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", RowBox[List["1", "-", "b", "+", "\[Alpha]"]], ",", RowBox[List["1", "-", "a", "+", "\[Alpha]"]], ",", "z"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "\[Alpha]"]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]]], "\[Pi]", " ", SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "-", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List["1", "+", "a", "-", "b"]], ",", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["1", "+", "\[Alpha]", "-", "b"]], "]"]], ">", "0"]]]]]]]]

 MathML Form

 0 t α - 1 - t z U TagBox["U", HypergeometricU] ( a , b , t ) t π 2 ( cot ( π ( b - α ) ) + cot ( π α ) ) csc ( b π ) Γ ( a - α ) Γ ( a ) Γ ( a - b + 1 ) Γ ( 1 - α ) Γ ( b - α ) 2 F 1 ( α , - b + α + 1 ; - a + α + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["\[Alpha]", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "b"]], "+", "\[Alpha]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "a"]], "+", "\[Alpha]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - π z a - α csc ( π ( a - α ) ) Γ ( a - α + 1 ) 2 F 1 ( a , a - b + 1 ; a - α + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "b", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "-", "\[Alpha]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; Re ( z ) > 0 Re ( α ) > 0 Re ( α - b + 1 ) > 0 Condition t 0 DirectedInfinity 1 t α -1 -1 t z HypergeometricU a b t 2 b -1 α α b Gamma a -1 α Gamma a Gamma a -1 b 1 Gamma 1 -1 α Gamma b -1 α -1 Hypergeometric2F1 α -1 b α 1 -1 a α 1 z -1 z a -1 α a -1 α Gamma a -1 α 1 -1 Hypergeometric2F1 a a -1 b 1 a -1 α 1 z z 0 α 0 α -1 b 1 0 [/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["-", "t_"]], " ", "z_"]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List["a_", ",", "b_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["b", "-", "\[Alpha]"]], ")"]]]], "]"]], "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Alpha]"]], "]"]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", RowBox[List["1", "-", "b", "+", "\[Alpha]"]], ",", RowBox[List["1", "-", "a", "+", "\[Alpha]"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "\[Alpha]"]], "]"]]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "-", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List["1", "+", "a", "-", "b"]], ",", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["1", "+", "\[Alpha]", "-", "b"]], "]"]], ">", "0"]]]]]]]]]]

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

 2002-12-18