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=25/8, b>=a > For fixed z and a=25/8, b=39/8





http://functions.wolfram.com/07.23.03.c9sm.01









  


  








Input Form





Hypergeometric2F1[25/8, 39/8, -(7/2), -z] == (1/(1277696 (1 + z)^(23/2))) ((-Sqrt[1 + z]) (-1277696 - 19655984 z - 159003991 z^2 - 1029019425 z^3 - 10264037132 z^4 + 34728929502 z^5 - 15167337279 z^6 + 569997987 z^7 + 13307346 z^8) Cosh[ArcSinh[Sqrt[z]]/4] + 2 Sqrt[z] (-159712 - 2506908 z - 20622099 z^2 - 134524562 z^3 - 39793366321 z^4 + 62757751752 z^5 - 16940661093 z^6 + 288325830 z^7 + 6653673 z^8) Sinh[ArcSinh[Sqrt[z]]/4])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["25", "8"], ",", FractionBox["39", "8"], ",", RowBox[List["-", FractionBox["7", "2"]]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["1277696", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["23", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["1", "+", "z"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1277696"]], "-", RowBox[List["19655984", " ", "z"]], "-", RowBox[List["159003991", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1029019425", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["10264037132", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["34728929502", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["15167337279", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["569997987", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["13307346", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["ArcSinh", "[", SqrtBox["z"], "]"]], "4"], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "159712"]], "-", RowBox[List["2506908", " ", "z"]], "-", RowBox[List["20622099", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["134524562", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["39793366321", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["62757751752", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["16940661093", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["288325830", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["6653673", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["ArcSinh", "[", 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> <mfrac> <mn> 25 </mn> <mn> 8 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 39 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </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[FractionBox[&quot;25&quot;, &quot;8&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;39&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;7&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[RowBox[List[&quot;-&quot;, &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> 1277696 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 23 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6653673 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 288325830 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 16940661093 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 62757751752 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 39793366321 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 134524562 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 20622099 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2506908 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 159712 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </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> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 13307346 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 569997987 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15167337279 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 34728929502 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10264037132 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1029019425 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 159003991 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 19655984 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1277696 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </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> <cn type='rational'> 25 <sep /> 8 </cn> <cn type='rational'> 39 <sep /> 8 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1277696 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 23 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6653673 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 288325830 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16940661093 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 62757751752 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 39793366321 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 134524562 </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'> 20622099 </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'> 2506908 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -159712 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <arcsinh /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 13307346 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 569997987 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15167337279 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 34728929502 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10264037132 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1029019425 </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'> 159003991 </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'> 19655984 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1277696 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <arcsinh /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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[FractionBox["25", "8"], ",", FractionBox["39", "8"], ",", RowBox[List["-", FractionBox["7", "2"]]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["1", "+", "z"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1277696"]], "-", RowBox[List["19655984", " ", "z"]], "-", RowBox[List["159003991", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1029019425", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["10264037132", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["34728929502", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["15167337279", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["569997987", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["13307346", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["ArcSinh", "[", SqrtBox["z"], "]"]], "4"], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "159712"]], "-", RowBox[List["2506908", " ", "z"]], "-", RowBox[List["20622099", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["134524562", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["39793366321", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["62757751752", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["16940661093", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["288325830", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["6653673", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["ArcSinh", "[", SqrtBox["z"], "]"]], "4"], "]"]]]]]], RowBox[List["1277696", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["23", "/", "2"]]]]]]]]]]


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





2007-05-02