Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











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 for products of 2F~1 with algebraic arguments





http://functions.wolfram.com/07.24.26.0112.01









  


  








Input Form





Hypergeometric2F1Regularized[a, a + 1/2, c, (1 - Sqrt[1 + z])/2] Hypergeometric2F1Regularized[a, a + 1/2, 2 a - c + 3/2, (1 - Sqrt[1 + z])/2] == (2^(-(3/2) + 4 a)/(Pi Gamma[2 a])) MeijerG[{{1/4 - a, 1/2 - a, 3/4 - a, 1 - a}, {}}, {{0}, {1 - c, -2 a + 1/2, -2 a + c - 1/2}}, z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", FractionBox["1", "2"]]], ",", "c", ",", FractionBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], "2"]]], "]"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["2", "a"]], "-", "c", "+", FractionBox["3", "2"]]], ",", FractionBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], "2"]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", RowBox[List["4", " ", "a"]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "a"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", "a"]], ",", RowBox[List[FractionBox["1", "2"], "-", "a"]], ",", RowBox[List[FractionBox["3", "4"], "-", "a"]], ",", RowBox[List["1", "-", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "c"]], ",", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "a"]], "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "a"]], "+", "c", "-", FractionBox["1", "2"]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;c&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SqrtBox[RowBox[List[&quot;1&quot;, &quot;+&quot;, &quot;z&quot;]]]]], &quot;2&quot;], Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]], &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SqrtBox[RowBox[List[&quot;1&quot;, &quot;+&quot;, &quot;z&quot;]]]]], &quot;2&quot;], Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> a </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> a </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> a </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> c </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </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;1&quot;, &quot;,&quot;, &quot;4&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[FractionBox[&quot;1&quot;, &quot;4&quot;], &quot;-&quot;, &quot;a&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, &quot;a&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;3&quot;, &quot;4&quot;], &quot;-&quot;, &quot;a&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;a&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;, &quot;c&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]], &quot;+&quot;, &quot;c&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <apply> <plus /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <apply> <plus /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> <ci> c </ci> </apply> </list> </list> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", RowBox[List["a_", "+", FractionBox["1", "2"]]], ",", "c_", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", RowBox[List["a_", "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "a_"]], "-", "c_", "+", FractionBox["3", "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", RowBox[List["4", " ", "a"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", "a"]], ",", RowBox[List[FractionBox["1", "2"], "-", "a"]], ",", RowBox[List[FractionBox["3", "4"], "-", "a"]], ",", RowBox[List["1", "-", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "c"]], ",", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", "c", "-", FractionBox["1", "2"]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "a"]], "]"]]]]]]]]]


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





2001-10-29