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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 > Generalized cases involving algebraic functions and 2F1with algebraic arguments





http://functions.wolfram.com/07.24.26.0261.01









  


  










Input Form





(z + Sqrt[1 + z^2])^(1 - c) Hypergeometric2F1Regularized[a, b, c, (z - Sqrt[1 + z^2])/(2 z)] Hypergeometric2F1[1 + a - c, 1 + b - c, 1 + a + b - c, (z - Sqrt[1 + z^2])/(2 z)] == ((2^(a + b - c) Gamma[1 + a + b - c])/(Sqrt[Pi] Gamma[a] Gamma[b])) MeijerG[{{(3 - c)/2}, {a + b + (1 - c)/2, a + b + (3 - 3 c)/2, (1 + c)/2}}, {{a + (1 - c)/2, b + (1 - c)/2, (1/2) (1 + a + b - c), 1 + (1/2) (a + b - c)}, {}}, z, 1/2] /; Re[z] > 0










Standard Form





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MathML Form







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</list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2001-10-29