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http://functions.wolfram.com/07.23.21.0007.01
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Integrate[(1 - z)^(\[Alpha] - 1) Hypergeometric2F1[a, b, c, z], z] ==
-((Gamma[c] MeijerG[{{1, 1 - a + \[Alpha], 1 - b + \[Alpha]}, {}},
{{\[Alpha], -a - b + c + \[Alpha]}, {0}}, 1 - z])/
(Gamma[a] Gamma[b] Gamma[-a + c] Gamma[-b + c]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "a", "+", "\[Alpha]"]], ",", RowBox[List["1", "-", "b", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]]], ")"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", 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] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> α </mi> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> α </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["1", "-", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "+", "\[Alpha]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "b"]], "+", "\[Alpha]", "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["\[Alpha]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c", "+", "\[Alpha]"]], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </list> <list /> </list> <list> <list> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <ci> α </ci> </apply> </list> <list> <cn type='integer'> 0 </cn> </list> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z_"]], ")"]], RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "a", "+", "\[Alpha]"]], ",", RowBox[List["1", "-", "b", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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HypergeometricPFQ[{},{},z] | HypergeometricPFQ[{},{b},z] | HypergeometricPFQ[{a},{},z] | HypergeometricPFQ[{a},{b},z] | HypergeometricPFQ[{a1},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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