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variants of this functions
BetaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z1,z2,a,b] > Series representations > Generalized power series > Expansions at {z1,z2}=={0,0}





http://functions.wolfram.com/06.22.06.0001.02









  


  










Input Form





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










Standard Form





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










MathML Form







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</mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> a </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <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> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mi> a </mi> </msubsup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> a </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> a </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <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> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </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> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <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> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Rule </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </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[FractionBox[RowBox[List[SubsuperscriptBox["zz", "2", "a"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "a"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SubscriptBox["zz", "2"]]], RowBox[List["1", "+", "a"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], " ", SubsuperscriptBox["zz", "2", "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]], "-", FractionBox[RowBox[List[SubsuperscriptBox["zz", "1", "a"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "a"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SubscriptBox["zz", "1"]]], RowBox[List["1", "+", "a"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], " ", SubsuperscriptBox["zz", "1", "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["zz", "1"], "\[Rule]", "0"]], ")"]], "&&", RowBox[List["(", RowBox[List[SubscriptBox["zz", "2"], "\[Rule]", "0"]], ")"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]










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





2001-10-29





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