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
Hypergeometric2F1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Specific values > Values at other z > Values at z==1221/2-16





http://functions.wolfram.com/07.23.03.0075.01









  


  








Input Form





Hypergeometric2F1[a, 2 a - 1/2, 4 a - 1, 12 Sqrt[2] - 16] == ((2 + Sqrt[2])^(2 a)/2^(4 a - 1)) Sqrt[Pi] (Gamma[2 a]/(Gamma[(a + 1)/2] Gamma[(3 a)/2]))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List[RowBox[List["2", "a"]], "-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["4", "a"]], "-", "1"]], ",", RowBox[List[RowBox[List["12", SqrtBox["2"]]], "-", "16"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", SqrtBox["2"]]], ")"]], RowBox[List["2", "a"]]], SuperscriptBox["2", RowBox[List[RowBox[List["4", "a"]], "-", "1"]]]], SqrtBox["\[Pi]"], FractionBox[RowBox[List["Gamma", "[", RowBox[List["2", "a"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["a", "+", "1"]], "2"], "]"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "a"]], "2"], "]"]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> - </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]], &quot;-&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;a&quot;]], &quot;-&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[RowBox[List[&quot;12&quot;, &quot; &quot;, SqrtBox[&quot;2&quot;]]], &quot;-&quot;, &quot;16&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <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> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -16 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", RowBox[List[RowBox[List["2", " ", "a_"]], "-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["4", " ", "a_"]], "-", "1"]], ",", RowBox[List[RowBox[List["12", " ", SqrtBox["2"]]], "-", "16"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", SqrtBox["2"]]], ")"]], RowBox[List["2", " ", "a"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "a"]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["4", " ", "a"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["a", "+", "1"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", " ", "a"]], "2"], "]"]]]], ")"]]]]]]]]]


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





2001-10-29