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variants of this functions
BetaRegularized




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.22.21.0005.01









  


  








Input Form





Integrate[BetaRegularized[Subscript[z, 1], \[ScriptA] Subscript[z, 2], a, b], Subscript[z, 2]] == Subscript[z, 2] BetaRegularized[Subscript[z, 1], \[ScriptA] Subscript[z, 2], a, b] - (a/(\[ScriptA] (a + b))) BetaRegularized[\[ScriptA] Subscript[z, 2], a + 1, b]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", RowBox[List["\[ScriptA]", " ", SubscriptBox["z", "2"]]], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z", "2"]]]]]]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["z", "2"], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", RowBox[List["\[ScriptA]", " ", 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", "2"]]], ",", RowBox[List["a", "+", "1"]], ",", "b"]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mrow> <mi> &#119990; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mrow> <mi> &#119990; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> a </mi> <mrow> <mi> &#119990; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mrow> <mi> &#119990; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <apply> <ci> BetaRegularized </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> &#119990; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> BetaRegularized </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> &#119990; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> &#119990; </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> &#119990; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", RowBox[List["\[ScriptA]_", " ", SubscriptBox["z_", "2"]]], ",", "a_", ",", "b_"]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z_", "2"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["zz", "2"], " ", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", RowBox[List["\[ScriptA]", " ", SubscriptBox["zz", "2"]]], ",", "a", ",", "b"]], "]"]]]], "-", FractionBox[RowBox[List["a", " ", RowBox[List["BetaRegularized", "[", RowBox[List[RowBox[List["\[ScriptA]", " ", SubscriptBox["zz", "2"]]], ",", RowBox[List["a", "+", "1"]], ",", "b"]], "]"]]]], RowBox[List["\[ScriptA]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]


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





2001-10-29