Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
HypergeometricPFQRegularized






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] > Specific values > Specialized values > Case 7F~6





http://functions.wolfram.com/07.32.03.0142.01









  


  










Input Form





HypergeometricPFQRegularized[{1, 1, 1, d, d, 2 - d, 2 - d}, {2, 2, 1 + d, 1 + d, 3 - d, 3 - d}, 1] == (1/(24 (-1 + d)^8 Pi^2)) (-12 + 8 Pi^2 - 16 d Pi^2 + 8 d^2 Pi^2 - 2 (-6 + (-1 + d)^2 Pi^2) Cos[2 d Pi] + 9 (-1 + d) Pi Sin[2 d Pi])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", "1", ",", "d", ",", "d", ",", RowBox[List["2", "-", "d"]], ",", RowBox[List["2", "-", "d"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2", ",", RowBox[List["1", "+", "d"]], ",", RowBox[List["1", "+", "d"]], ",", RowBox[List["3", "-", "d"]], ",", RowBox[List["3", "-", "d"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "8"], " ", SuperscriptBox["\[Pi]", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["-", "12"]], "+", RowBox[List["8", " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["16", " ", "d", " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["d", "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "d", " ", "\[Pi]"]], "]"]]]], "+", RowBox[List["9", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "d", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]]]]










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> 7 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 6 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> d </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> d </mi> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;7&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;6&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;d&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;d&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;d&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;d&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;2&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;2&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;3&quot;, &quot;-&quot;, &quot;d&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;3&quot;, &quot;-&quot;, &quot;d&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 12 </mn> </mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> <ci> d </ci> <ci> d </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </list> <list> <cn type='integer'> 2 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </list> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> d </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <pi /> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <pi /> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <pi /> </apply> </apply> </apply> <cn type='integer'> -12 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </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["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", "1", ",", "d_", ",", "d_", ",", RowBox[List["2", "-", "d_"]], ",", RowBox[List["2", "-", "d_"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2", ",", RowBox[List["1", "+", "d_"]], ",", RowBox[List["1", "+", "d_"]], ",", RowBox[List["3", "-", "d_"]], ",", RowBox[List["3", "-", "d_"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", "12"]], "+", RowBox[List["8", " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["16", " ", "d", " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["d", "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "d", " ", "\[Pi]"]], "]"]]]], "+", RowBox[List["9", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "d", " ", "\[Pi]"]], "]"]]]]]], RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "8"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.