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variants of this functions
BetaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z,a,b] > Series representations > Generalized power series > Expansions at z==1 > For the function itself > Generic formulas for main term





http://functions.wolfram.com/06.21.06.0044.01









  


  










Input Form





BetaRegularized[z, a, b] \[Proportional] Piecewise[{{0, Element[-b, Integers] && -b >= 0}, {1, Re[b] > 0}, {-((1 - z)^b/(b Beta[a, b])), Re[b] < 0}}, 1 - (1 - z)^b/(b Beta[a, b])] /; (z -> 1)










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </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> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </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> </mrow> </mfrac> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> BetaRegularized </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> <piecewise> <piece> <cn type='integer'> 0 </cn> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> &#8469; </ci> </apply> </piece> <piece> <cn type='integer'> 1 </cn> <apply> <gt /> <apply> <real /> <ci> b </ci> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <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> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <real /> <ci> b </ci> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <otherwise> <apply> <plus /> <cn type='integer'> 1 </cn> <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> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List["0", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "b"]], "\[GreaterEqual]", "0"]]]]], List["1", RowBox[List[RowBox[List["Re", "[", "b", "]"]], ">", "0"]]], List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], RowBox[List["b", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]], RowBox[List[RowBox[List["Re", "[", "b", "]"]], "<", "0"]]], List[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], RowBox[List["b", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]]]]]]]]










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





2007-05-02





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