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http://functions.wolfram.com/06.22.21.0001.01
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Integrate[BetaRegularized[\[ScriptA] Subscript[z, 1], Subscript[z, 2], a, b],
Subscript[z, 1]] ==
Subscript[z, 1] BetaRegularized[\[ScriptA] Subscript[z, 1], Subscript[z, 2],
a, b] + (a/(\[ScriptA] (a + b))) BetaRegularized[
\[ScriptA] Subscript[z, 1], 1 + a, b]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]", " ", SubscriptBox["z", "1"]]], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z", "1"]]]]]]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["z", "1"], " ", RowBox[List["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]", " ", SubscriptBox["z", "1"]]], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["a", " "]], RowBox[List["\[ScriptA]", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]], RowBox[List["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]", " ", SubscriptBox["z", "1"]]], ",", RowBox[List["1", "+", "a"]], ",", "b"]], "]"]]]]]]]]]]
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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> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> 𝒶 </mi> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mi> a </mi> <mrow> <mi> 𝒶 </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mrow> <mi> 𝒶 </mi> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> 𝒶 </mi> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <apply> <ci> BetaRegularized </ci> <apply> <times /> <ci> 𝒶 </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> 𝒶 </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BetaRegularized </ci> <apply> <times /> <ci> 𝒶 </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> b </ci> </apply> </apply> <apply> <times /> <apply> <ci> BetaRegularized </ci> <apply> <times /> <ci> 𝒶 </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> <ci> b </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </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["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]_", " ", SubscriptBox["z_", "1"]]], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z_", "1"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["zz", "1"], " ", RowBox[List["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]", " ", SubscriptBox["zz", "1"]]], ",", SubscriptBox["zz", "2"], ",", "a", ",", "b"]], "]"]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]", " ", SubscriptBox["zz", "1"]]], ",", RowBox[List["1", "+", "a"]], ",", "b"]], "]"]]]], RowBox[List["\[ScriptA]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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