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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==infinity





http://functions.wolfram.com/06.21.06.0017.02









  


  










Input Form





BetaRegularized[z, a, b] \[Proportional] ((z^a (-z)^(b - 1))/((a + b - 1) Beta[a, b])) (1 + O[1/z]) - ((Sin[a Pi]/Pi) z^a (Log[-z] - PolyGamma[a] + PolyGamma[a + b]))/(-z)^a /; Element[a + b - 1, Integers] && a + b - 1 > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], " "]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]], "\[Pi]"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox["z", "a"], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], ">", "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> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <msup> <mi> z </mi> <mi> a </mi> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;z&quot;, &quot;a&quot;], Pochhammer] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </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 /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <sin /> <apply> <times /> <ci> a </ci> <pi /> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <apply> <ci> Pochhammer </ci> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <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> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox["z", "a"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]]]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], ">", "0"]]]]]]]]]]










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





2001-10-29