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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 algebraic functions and 2F1with algebraic arguments





http://functions.wolfram.com/07.24.26.0188.01









  


  








Input Form





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








Standard Form





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MathML Form







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</mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> c </mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;4&quot;, &quot;,&quot;, &quot;4&quot;]], RowBox[List[&quot;4&quot;, &quot;,&quot;, &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, FractionBox[&quot;c&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;a&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]], &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, &quot;c&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;b&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> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </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["2", "+", "z_", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z_"]]]]]]], ")"]], RowBox[List["a_", "+", "b_", "-", "c_"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]]]], "z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "a_"]], "+", "c_"]], ",", RowBox[List[RowBox[List["-", "b_"]], "+", "c_"]], ",", "c_", ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]]]], "z_"]]]]], "]"]]]], "]"]], "\[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["1", "+", "a", "+", "b", "-", "c"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "b", "-", FractionBox["c", "2"]]], ",", RowBox[List[FractionBox[RowBox[List["1", "-", "c"]], "2"], "+", "a", "+", "b"]], ",", RowBox[List["a", "+", "b"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", "c"]], ",", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "-", "c"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], 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["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29