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=-11/2, b>=a > For fixed z and a=-11/2, b=-1/2





http://functions.wolfram.com/07.23.03.0842.01









  


  










Input Form





Hypergeometric2F1[-(11/2), -(1/2), 5, z] == (1/(72747675 Pi z^4)) (256 ((-176 + z (2464 + z (-18051 + z (112255 + z (1070538 + z (179134 + z (-44191 + z (10259 + 8 z (-205 + 16 z))))))))) EllipticE[z] - 4 (-1 + z) (44 + z (-594 + z (4224 + z (-26059 + z (-137840 + z (-4556 + z (1116 + z (-191 + 16 z)))))))) EllipticK[z]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", "5", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["72747675", " ", "\[Pi]", " ", SuperscriptBox["z", "4"]]]], RowBox[List["(", RowBox[List["256", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "176"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["2464", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "18051"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["112255", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1070538", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["179134", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "44191"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["10259", "+", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "205"]], "+", RowBox[List["16", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "z", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["44", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "594"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["4224", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "26059"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "137840"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4556"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1116", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "191"]], "+", RowBox[List["16", " ", "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> 11 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </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;11&quot;, &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&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> 72747675 </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> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 205 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 10259 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 44191 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 179134 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1070538 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 112255 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 18051 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2464 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 176 </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> 4 </mn> <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> <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> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 191 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1116 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 4556 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 137840 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 26059 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4224 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 594 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 44 </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'> 11 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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'> 72747675 </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 /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> z </ci> </apply> <cn type='integer'> -205 </cn> </apply> </apply> <cn type='integer'> 10259 </cn> </apply> </apply> <cn type='integer'> -44191 </cn> </apply> </apply> <cn type='integer'> 179134 </cn> </apply> </apply> <cn type='integer'> 1070538 </cn> </apply> </apply> <cn type='integer'> 112255 </cn> </apply> </apply> <cn type='integer'> -18051 </cn> </apply> </apply> <cn type='integer'> 2464 </cn> </apply> </apply> <cn type='integer'> -176 </cn> </apply> <apply> <times /> <exponentiale /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <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 /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> z </ci> </apply> <cn type='integer'> -191 </cn> </apply> </apply> <cn type='integer'> 1116 </cn> </apply> </apply> <cn type='integer'> -4556 </cn> </apply> </apply> <cn type='integer'> -137840 </cn> </apply> </apply> <cn type='integer'> -26059 </cn> </apply> </apply> <cn type='integer'> 4224 </cn> </apply> </apply> <cn type='integer'> -594 </cn> </apply> </apply> <cn type='integer'> 44 </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["11", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", "5", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["256", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "176"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["2464", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "18051"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["112255", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1070538", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["179134", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "44191"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["10259", "+", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "205"]], "+", RowBox[List["16", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "z", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["44", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "594"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["4224", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "26059"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "137840"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4556"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1116", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "191"]], "+", RowBox[List["16", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["72747675", " ", "\[Pi]", " ", SuperscriptBox["z", "4"]]]]]]]]










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





2007-05-02