Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











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 > General case





http://functions.wolfram.com/06.21.06.0042.01









  


  








Input Form





BetaRegularized[z, a, b] \[Proportional] 1 - ((1 - z)^b/(b Beta[a, b])) (1 + ((a - 1) b (z - 1))/(1 + b) + ((a - 1) (a - 2) b (z - 1)^2)/(2 (2 + b)) + O[(z - 1)^3]) /; !(Element[-b, Integers] && -b >= 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"]]], RowBox[List["b", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", "b", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "+", "b"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", "b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["-", "b"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["-", "b"]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]








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> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <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> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </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> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <ci> b </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -2 </cn> </apply> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> &#8469; </ci> </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["1", "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", "b", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "+", "b"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", "b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]], "3"]]], ")"]]]], RowBox[List["b", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "b"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]


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





2007-05-02