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variants of this functions
Beta






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z,a,b] > Identities > Functional identities > Major general cases





http://functions.wolfram.com/06.19.17.0009.01









  


  










Input Form





Beta[z, a, b] == (z^a ((1/z)^(a + b) (-z)^(a + b) Beta[1/z, 1 - a - b, b] + Beta[a, 1 - a - b]))/(-z)^a /; !(Element[a + b, Integers] && a + b > 0) && !IntervalMemberQ[Interval[{0, 1}], z]










Standard Form





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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> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <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> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </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> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> &#8713; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <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> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </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> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> &#8713; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox["z", "a"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["a", "+", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["a", "+", "b"]]], " ", RowBox[List["Beta", "[", RowBox[List[FractionBox["1", "z"], ",", RowBox[List["1", "-", "a", "-", "b"]], ",", "b"]], "]"]]]], "+", RowBox[List["Beta", "[", RowBox[List["a", ",", RowBox[List["1", "-", "a", "-", "b"]]]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["a", "+", "b"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], ">", "0"]]]], ")"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]










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





2001-10-29