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 rational parameters with denominators 8 and fixed z and a<0 > For fixed z and a=-13/8, b>=a > For fixed z and a=-13/8, b=45/8





http://functions.wolfram.com/07.23.03.bqtw.01









  


  








Input Form





Hypergeometric2F1[-(13/8), 45/8, -(5/2), z] == (1/(68672 (1 - z)^(13/2))) (4 Sqrt[1 - z] (17168 - 45066 z - 22533 z^2 - 75110 z^3 + 487143 z^4 - 868998 z^5 + 750516 z^6 - 325584 z^7 + 57120 z^8) Cos[(3 ArcSin[Sqrt[z]])/4] + Sqrt[z] (51504 - 142709 z - 52577 z^2 - 983231 z^3 + 3383781 z^4 - 4828512 z^5 + 3624672 z^6 - 1416576 z^7 + 228480 z^8) Sin[(3 ArcSin[Sqrt[z]])/4])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["13", "8"]]], ",", FractionBox["45", "8"], ",", RowBox[List["-", FractionBox["5", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["68672", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["13", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["17168", "-", RowBox[List["45066", " ", "z"]], "-", RowBox[List["22533", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["75110", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["487143", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["868998", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["750516", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["325584", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["57120", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]], "4"], "]"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List["51504", "-", RowBox[List["142709", " ", "z"]], "-", RowBox[List["52577", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["983231", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["3383781", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["4828512", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["3624672", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["1416576", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["228480", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]], "4"], "]"]]]]]], ")"]]]]]]]]








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> 13 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 45 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> <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;13&quot;, &quot;8&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;45&quot;, &quot;8&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;5&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 68672 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 13 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 57120 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 325584 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 750516 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 868998 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 487143 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 75110 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 22533 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 45066 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 17168 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 228480 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1416576 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3624672 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4828512 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3383781 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 983231 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 52577 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 142709 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 51504 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 13 <sep /> 8 </cn> </apply> <cn type='rational'> 45 <sep /> 8 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 68672 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 57120 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 325584 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 750516 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 868998 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 487143 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 75110 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 22533 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 45066 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 17168 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 228480 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1416576 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3624672 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4828512 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3383781 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 983231 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 52577 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 142709 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 51504 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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["13", "8"]]], ",", FractionBox["45", "8"], ",", RowBox[List["-", FractionBox["5", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["17168", "-", RowBox[List["45066", " ", "z"]], "-", RowBox[List["22533", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["75110", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["487143", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["868998", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["750516", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["325584", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["57120", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]], "4"], "]"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List["51504", "-", RowBox[List["142709", " ", "z"]], "-", RowBox[List["52577", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["983231", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["3383781", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["4828512", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["3624672", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["1416576", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["228480", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]], "4"], "]"]]]]]], RowBox[List["68672", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["13", "/", "2"]]]]]]]]]]


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





2007-05-02