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 > For integer and half-integer parameters and fixed z > For fixed z and a=-9/2, b>=a > For fixed z and a=-9/2, b=5/2





http://functions.wolfram.com/07.23.03.1369.01









  


  








Input Form





Hypergeometric2F1[-(9/2), 5/2, 5, z] == (1/(765765 Pi z^4)) (256 ( (-336 + z (1344 + z (-1029 + z (-1617 + z (12001 + z (-19739 + 32 z (489 + 4 z (-49 + 8 z)))))))) EllipticE[z] - 8 (-2 + z) (-1 + z) (-21 + z (63 + z^2 (-105 + z (271 + 16 z (-13 + 4 z))))) EllipticK[z]))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["9", "2"]]], ",", FractionBox["5", "2"], ",", "5", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["765765", " ", "\[Pi]", " ", SuperscriptBox["z", "4"]]]], RowBox[List["(", RowBox[List["256", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "336"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1344", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1029"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1617"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["12001", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "19739"]], "+", RowBox[List["32", " ", "z", " ", RowBox[List["(", RowBox[List["489", "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "49"]], "+", RowBox[List["8", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "z", "]"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "21"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["63", "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "105"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["271", "+", RowBox[List["16", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "13"]], "+", RowBox[List["4", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "z", "]"]]]]]], ")"]]]], ")"]]]]]]]]








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> <mrow> <mo> - </mo> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mn> 5 </mn> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;2&quot;], SubscriptBox[&quot;F&quot;, &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;9&quot;, &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;5&quot;, &quot;2&quot;], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;5&quot;, Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 765765 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 256 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 49 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 489 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 19739 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 12001 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1617 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1029 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1344 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 336 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 13 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 271 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 105 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 63 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 21 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 9 <sep /> 2 </cn> </apply> <cn type='rational'> 5 <sep /> 2 </cn> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 765765 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> <cn type='integer'> -49 </cn> </apply> </apply> <cn type='integer'> 489 </cn> </apply> </apply> <cn type='integer'> -19739 </cn> </apply> </apply> <cn type='integer'> 12001 </cn> </apply> </apply> <cn type='integer'> -1617 </cn> </apply> </apply> <cn type='integer'> -1029 </cn> </apply> </apply> <cn type='integer'> 1344 </cn> </apply> </apply> <cn type='integer'> -336 </cn> </apply> <apply> <times /> <exponentiale /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> -13 </cn> </apply> </apply> <cn type='integer'> 271 </cn> </apply> </apply> <cn type='integer'> -105 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 63 </cn> </apply> </apply> <cn type='integer'> -21 </cn> </apply> <apply> <times /> <ci> K </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["9", "2"]]], ",", FractionBox["5", "2"], ",", "5", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["256", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "336"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1344", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1029"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1617"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["12001", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "19739"]], "+", RowBox[List["32", " ", "z", " ", RowBox[List["(", RowBox[List["489", "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "49"]], "+", RowBox[List["8", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "z", "]"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "21"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["63", "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "105"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["271", "+", RowBox[List["16", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "13"]], "+", RowBox[List["4", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["765765", " ", "\[Pi]", " ", SuperscriptBox["z", "4"]]]]]]]]


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





2007-05-02