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http://functions.wolfram.com/06.18.21.0005.01
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Integrate[Beta[a, b], b] ==
Log[b] + Sum[(Pochhammer[1 - a, k + 1]/(k + 1)!) Log[1 + b/(1 + k)],
{k, 0, Infinity}] /; Re[a] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "b"]]]]]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", "b", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["k", "+", "1"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]], RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["b", RowBox[List["1", "+", "k"]]]]], "]"]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]]]]]]
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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> <mrow> <mo> ∫ </mo> <mrow> <mrow> <semantics> <mi> Β </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> b </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], RowBox[List["k", "+", "1"]]], Pochhammer] </annotation> </semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> b </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> b </ci> </bvar> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <ln /> <ci> b </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["Beta", "[", RowBox[List["a_", ",", "b_"]], "]"]], RowBox[List["\[DifferentialD]", "b_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Log", "[", "b", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["k", "+", "1"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["b", RowBox[List["1", "+", "k"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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