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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z1,z2,a,b] > Integration > Indefinite integration > Involving only one direct function with respect to a





http://functions.wolfram.com/06.20.21.0009.01









  


  








Input Form





Integrate[Beta[Subscript[z, 1], Subscript[z, 2], a, b], a] == Gamma[0, (-a) Log[Subscript[z, 1]], (-a) Log[Subscript[z, 2]]] + Log[(-a) Log[Subscript[z, 1]]] - Log[(-a) Log[Subscript[z, 2]]] + (1 - b) Sum[((Pochhammer[2 - b, k] (Subscript[z, 2]^(k + a + 1) - Subscript[z, 1]^(k + a + 1)))/(k + 1)!) Log[1 + a/(1 + k)], {k, 0, Infinity}] /; Abs[Subscript[z, 1]] < 1 && Abs[Subscript[z, 2]] < 1 && !(Element[-a, Integers] && -a >= 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "a"]]]]]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]], ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["z", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["a", RowBox[List["1", "+", "k"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "1"], "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]








MathML Form







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</mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> a </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]], RowBox[List["\[DifferentialD]", "a_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]], ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["zz", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["a", RowBox[List["1", "+", "k"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "1"], "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]


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





2001-10-29