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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/07.24.26.0256.01









  


  








Input Form





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








Standard Form





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MathML Form







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&quot;, &quot;z&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]], MeijerG], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;a&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;+&quot;, &quot;a&quot;, &quot;-&quot;, &quot;c&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, &quot;c&quot;]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;c&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;c&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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


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





2001-10-29