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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 one direct function and elementary functions with respect to z1 > Involving power function





http://functions.wolfram.com/06.22.21.0008.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SubsuperscriptBox["z", "2", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z", "2"]]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["z", "2", "\[Alpha]"], " "]], "\[Alpha]"], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", RowBox[List[" ", "1"]]], ",", SubscriptBox["z", RowBox[List[" ", "2"]]], ",", "a", ",", "b"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Alpha]"]], "]"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Alpha]"]], "]"]]]]], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", RowBox[List[" ", "2"]]], ",", RowBox[List["\[Alpha]", "+", "a"]], ",", "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> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <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> <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> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <msubsup> <mi> z </mi> <mn> 2 </mn> <mi> &#945; </mi> </msubsup> <mi> &#945; </mi> </mfrac> <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> <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> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#945; </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> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> &#945; </mi> </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> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BetaRegularized </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#945; </ci> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BetaRegularized </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> &#945; </ci> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#945; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BetaRegularized </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> &#945; </ci> </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[SubsuperscriptBox["z_", "2", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", SubscriptBox["z_", "2"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["zz", "2", "\[Alpha]"], " ", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"], ",", "a", ",", "b"]], "]"]]]], "\[Alpha]"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Alpha]"]], "]"]]]], ")"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "2"], ",", RowBox[List["\[Alpha]", "+", "a"]], ",", "b"]], "]"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Alpha]"]], "]"]]]]]]]]]]]


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





2001-10-29