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variants of this functions
InverseBetaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > InverseBetaRegularized[z,a,b] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/06.23.06.0001.01









  


  










Input Form





InverseBetaRegularized[z, a, b] \[Proportional] (a z Beta[a, b])^(1/a) + ((b - 1)/(1 + a)) (a z Beta[a, b])^(2/a) + (((b - 1) (-4 - a + a^2 + 5 b + 3 a b))/(2 (1 + a)^2 (2 + a))) (a ((-(z - 1)) Gamma[a + 1])^(1/a) z Beta[a, b])^(3/a) + O[z^(4/a)] /; a > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]], ")"]], RowBox[List["1", "/", "a"]]], "+", RowBox[List[FractionBox[RowBox[List["b", "-", "1", " "]], RowBox[List["1", "+", "a"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]], ")"]], RowBox[List["2", "/", "a"]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "-", "a", "+", SuperscriptBox["a", "2"], "+", RowBox[List["5", " ", "b"]], "+", RowBox[List["3", " ", "a", " ", "b"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]]], ")"]], RowBox[List["1", "/", "a"]]], "z", " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]], ")"]], RowBox[List["3", "/", "a"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["4", "/", "a"]]], "]"]]]]]], "/;", RowBox[List["a", ">", "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> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </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> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> <mtext> </mtext> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </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> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </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> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mrow> <mn> 4 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <ci> b </ci> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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