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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z,a,b] > Series representations > Generalized power series > Expansions at z==1 > For the function itself > Logarithmic cases





http://functions.wolfram.com/06.19.06.0044.01









  


  








Input Form





Beta[z, a, 0] \[Proportional] -Log[1 - z] + (-EulerGamma - PolyGamma[a] + (1 - a) (z - 1) + (1/4) (2 - a) (-1 + a) (z - 1)^2 + \[Ellipsis]) /; (z -> 1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "0"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", "a"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]]]]]]








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> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <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> </mrow> <mo> - </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </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> Beta </ci> <ci> z </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </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["Beta", "[", RowBox[List["z_", ",", "a_", ",", "0"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", "a"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]]]]]]]]


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





2007-05-02