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variants of this functions
Hypergeometric2F1Regularized




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1Regularized[a,b,c,z] > Representations through more general functions > Through Meijer G > Classical cases involving 2F1with algebraic arguments





http://functions.wolfram.com/07.24.26.0175.01









  


  








Input Form





Hypergeometric2F1Regularized[a, b, (1 + a + b)/2, (Sqrt[z] - Sqrt[1 + z])/ (2 Sqrt[z])] Hypergeometric2F1[1 - a, 1 - b, (3 - a - b)/2, (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] == ((Cos[(1/2) (a - b) Pi] Gamma[(3 - a - b)/2])/Pi^(3/2)) MeijerG[{{1}, {(1 + a + b)/2, (1/2) (3 - a - b)}}, {{1/2, (1/2) (1 + a - b), (1/2) (1 - a + b)}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a_", "+", "b_"]], ")"]]]], ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", "a_"]], ",", RowBox[List["1", "-", "b_"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a_", "-", "b_"]], ")"]]]], ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a", "-", "b"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a", "-", "b"]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "-", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", "b"]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29