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variants of this functions
BetaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z1,z2,a,b] > Identities > Recurrence identities > Consecutive neighbors





http://functions.wolfram.com/06.22.17.0002.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", RowBox[List["a", "-", "1"]], ",", "b"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "-", "1"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "1"]]], ")"]], "b"], " ", SubsuperscriptBox["z", "1", RowBox[List["a", "-", "1"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "2"]]], ")"]], "b"], " ", SubsuperscriptBox["z", "2", RowBox[List["a", "-", "1"]]]]]]], ")"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> <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> &#10869; </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> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <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> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <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> <plus /> <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> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"], ",", RowBox[List["a", "-", "1"]], ",", "b"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "-", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "1"]]], ")"]], "b"], " ", SubsuperscriptBox["zz", "1", RowBox[List["a", "-", "1"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "2"]]], ")"]], "b"], " ", SubsuperscriptBox["zz", "2", RowBox[List["a", "-", "1"]]]]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]]]]]]]]










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





2001-10-29