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http://functions.wolfram.com/06.19.26.0008.01
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Beta[z, a, b] == (Beta[1 - a - b, a] z^a)/(-z)^a +
(1/(a + b - 1)) (-z)^(b - 1) z^a HypergeometricPFQ[{1 - b, 1 - a - b},
{2 - a - b}, 1/z] /; !(Element[a + b, Integers] && a + b > 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List[RowBox[List["1", "-", "a", "-", "b"]], ",", "a"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox["z", "a"]]], "+", RowBox[List[FractionBox["1", RowBox[List["a", "+", "b", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]], " ", SuperscriptBox["z", "a"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", RowBox[List["1", "-", "a", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "a", "-", "b"]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["a", "+", "b"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b"]], ">", "0"]]]], "]"]]]]]]
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<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> Β </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> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ⁢ </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> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mi> Β </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ∉ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Beta </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <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> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> a </ci> </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> </apply> <apply> <notin /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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