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http://functions.wolfram.com/06.19.06.0059.01
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Beta[z, a, b] \[Proportional]
Piecewise[{{-((E^(I b Pi) z^(-1 + a + b))/(-1 + a + b)) +
(Gamma[b] Log[z])/(E^(I a Pi) (Gamma[1 - a] Gamma[a + b])),
Arg[z] <= 0}}, -(z^(-1 + a + b)/(E^(I b Pi) (-1 + a + b))) +
(E^(I a Pi) Gamma[b] Log[z])/(Gamma[1 - a] Gamma[a + b])] /;
Element[a + b - 1, Integers] && a + b - 1 > 0 && (Abs[z] -> Infinity)
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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> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> + </mo> <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> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> + </mo> <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> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ∉ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </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> <ci> b </ci> </apply> <piecewise> <piece> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> b </ci> <pi /> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> a </ci> <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> </apply> <apply> <leq /> <apply> <arg /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> a </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> <pi /> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> a </ci> <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> </apply> </otherwise> </piecewise> </apply> <apply> <and /> <apply> <notin /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]]]]], RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "\[Pi]"]]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]]]]], RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]], List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "\[Pi]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]]]]], RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Pi]"]]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], ">", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
