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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.21.0004.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) Hypergeometric2F1Regularized[a, b, c, -t])/ E^(d t), {t, 0, Infinity}] == ((Gamma[a - \[Alpha]] Gamma[b - \[Alpha]] Gamma[\[Alpha]])/ (Gamma[a] Gamma[b] Gamma[c - \[Alpha]])) HypergeometricPFQ[ {\[Alpha], 1 - c + \[Alpha]}, {1 - a + \[Alpha], 1 - b + \[Alpha]}, d] + ((Gamma[a - b] Gamma[\[Alpha] - b])/(Gamma[a] Gamma[c - b])) d^(b - \[Alpha]) HypergeometricPFQ[{b, b - c + 1}, {1 - a + b, b - \[Alpha] + 1}, d] + ((Gamma[b - a] Gamma[\[Alpha] - a])/(Gamma[b] Gamma[c - a])) d^(a - \[Alpha]) HypergeometricPFQ[{a, a - c + 1}, {a - b + 1, a - \[Alpha] + 1}, d] /; Re[\[Alpha]] > 0 && Re[d] > 0

 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["-", "d"]], " ", "t"]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", RowBox[List["-", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "\[Alpha]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", RowBox[List["1", "-", "c", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a", "+", "\[Alpha]"]], ",", RowBox[List["1", "-", "b", "+", "\[Alpha]"]]]], "}"]], ",", "d"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "-", "b"]], "]"]]]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "b"]], "]"]]]]], " ", SuperscriptBox["d", RowBox[List["b", "-", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", RowBox[List["b", "-", "c", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a", "+", "b"]], ",", RowBox[List["b", "-", "\[Alpha]", "+", "1"]]]], "}"]], ",", "d"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["b", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "-", "a"]], "]"]]]]]], RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]]]]], " ", SuperscriptBox["d", RowBox[List["a", "-", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", RowBox[List["a", "-", "c", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "-", "b", "+", "1"]], ",", RowBox[List["a", "-", "\[Alpha]", "+", "1"]]]], "}"]], ",", "d"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "d", "]"]], ">", "0"]]]]]]]]

 MathML Form

 0 t α - 1 - d t 2 F ~ 1 ( a , b ; c ; - t ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "t"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] t Γ ( a - α ) Γ ( b - α ) Γ ( α ) Γ ( a ) Γ ( b ) Γ ( c - α ) 2 F 2 ( α , 1 - c + α ; 1 - a + α , 1 - b + α ; d ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "c", "+", "\[Alpha]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "a", "+", "\[Alpha]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b", "+", "\[Alpha]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["d", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + Γ ( a - b ) Γ ( α - b ) Γ ( a ) Γ ( c - b ) d b - α 2 F 2 ( b , b - c + 1 ; 1 - a + b , b - α + 1 ; d ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "a", "+", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "\[Alpha]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["d", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + Γ ( b - a ) Γ ( α - a ) Γ ( b ) Γ ( c - a ) d a - α 2 F 2 ( a , a - c + 1 ; a - b + 1 , a - α + 1 ; d ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "c", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "-", "b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "\[Alpha]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["d", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; Re ( α ) > 0 Re ( d ) > 0 Condition t 0 t α -1 -1 d t Hypergeometric2F1Regularized a b c -1 t Gamma a -1 α Gamma b -1 α Gamma α Gamma a Gamma b Gamma c -1 α -1 HypergeometricPFQ α 1 -1 c α 1 -1 a α 1 -1 b α d Gamma a -1 b Gamma α -1 b Gamma a Gamma c -1 b -1 d b -1 α HypergeometricPFQ b b -1 c 1 1 -1 a b b -1 α 1 d Gamma b -1 a Gamma α -1 a Gamma b Gamma c -1 a -1 d a -1 α HypergeometricPFQ a a -1 c 1 a -1 b 1 a -1 α 1 d α 0 d 0 [/itex]

 Rule Form

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

 2001-10-29