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variants of this functions
BetaRegularized




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z1,z2,a,b] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/06.22.26.0006.01









  


  








Input Form





BetaRegularized[Subscript[z, 1], Subscript[z, 2], a, b] == Csc[Pi (a + b)] Sin[b Pi] (Subscript[z, 2]^a/(-Subscript[z, 2])^a - Subscript[z, 1]^a/(-Subscript[z, 1])^a) + (1/((a + b - 1) Beta[a, b])) ((-Subscript[z, 2])^(b - 1) Subscript[z, 2]^a HypergeometricPFQ[ {1 - b, 1 - a - b}, {2 - a - b}, 1/Subscript[z, 2]] - (-Subscript[z, 1])^(b - 1) Subscript[z, 1]^a HypergeometricPFQ[ {1 - b, 1 - a - b}, {2 - a - b}, 1/Subscript[z, 1]]) /; !(Element[a + b, Integers] && a + b > 0)








Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "2"]]], ")"]], RowBox[List["-", "a"]]], " ", SubsuperscriptBox["zz", "2", "a"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["-", "a"]]], " ", SubsuperscriptBox["zz", "1", "a"]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "2"]]], ")"]], RowBox[List["b", "-", "1"]]], " ", SubsuperscriptBox["zz", "2", "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", SubscriptBox["zz", "2"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["b", "-", "1"]]], " ", SubsuperscriptBox["zz", "1", "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", SubscriptBox["zz", "1"]]]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["a", "+", "b"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], ">", "0"]]]], ")"]]]]]]]]]]


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





2001-10-29