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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z1,z2,a,b] > General characteristics > Symmetries and periodicities > Mirror symmetry





http://functions.wolfram.com/06.20.04.0002.02









  


  








Input Form





Beta[Conjugate[Subscript[z, 1]], Conjugate[Subscript[z, 2]], Conjugate[a], Conjugate[b]] == Conjugate[Beta[Subscript[z, 1], Subscript[z, 2], a, b]] /; !IntervalMemberQ[Interval[{-Infinity, 0}], Subscript[z, 1]] && !IntervalMemberQ[Interval[{1, Infinity}], Subscript[z, 1]] && !IntervalMemberQ[Interval[{-Infinity, 0}], Subscript[z, 2]] && !IntervalMemberQ[Interval[{1, Infinity}], Subscript[z, 2]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List[RowBox[List["Conjugate", "[", SubscriptBox["z", "1"], "]"]], ",", RowBox[List["Conjugate", "[", SubscriptBox["z", "2"], "]"]], ",", RowBox[List["Conjugate", "[", "a", "]"]], ",", RowBox[List["Conjugate", "[", "b", "]"]]]], "]"]], "\[Equal]", RowBox[List["Conjugate", "[", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], "]"]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", SubscriptBox["z", "1"]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", SubscriptBox["z", "1"]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", SubscriptBox["z", "2"]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", SubscriptBox["z", "2"]]], "]"]], "]"]]]]]]]]








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 encoding='Mathematica'> TagBox[&quot;\[CapitalBeta]&quot;, BetaRegularized] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mover> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> _ </mo> </mover> <mo> , </mo> <mover> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> _ </mo> </mover> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mover> <mi> a </mi> <mo> _ </mo> </mover> <mo> , </mo> <mover> <mi> b </mi> <mo> _ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mover> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[CapitalBeta]&quot;, BetaRegularized] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[CapitalBeta]&quot;, BetaRegularized] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mover> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> _ </mo> </mover> <mo> , </mo> <mover> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> _ </mo> </mover> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mover> <mi> a </mi> <mo> _ </mo> </mover> <mo> , </mo> <mover> <mi> b </mi> <mo> _ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mover> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[CapitalBeta]&quot;, BetaRegularized] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </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[RowBox[List["Conjugate", "[", SubscriptBox["z_", "1"], "]"]], ",", RowBox[List["Conjugate", "[", SubscriptBox["z_", "2"], "]"]], ",", RowBox[List["Conjugate", "[", "a_", "]"]], ",", RowBox[List["Conjugate", "[", "b_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Conjugate", "[", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"], ",", "a", ",", "b"]], "]"]], "]"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", SubscriptBox["zz", "1"]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", SubscriptBox["zz", "1"]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", SubscriptBox["zz", "2"]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", SubscriptBox["zz", "2"]]], "]"]]]]]]]]]]]]


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





2001-10-29