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 4 and fixed z > For fixed z and a=-21/4, b>=a > For fixed z and a=-21/4, b=5





http://functions.wolfram.com/07.23.03.a8vz.01









  


  








Input Form





Hypergeometric2F1[-(21/4), 5, -(1/4), z] == (1/8388608) (8388608 + 880803840 z - 14327498850 z^2 + 57983184350 z^3 - 85053518550 z^4 + 41083292250 z^5 + 69615 z^(5/4) (13923 - 154700 z + 517650 z^2 - 669900 z^3 + 295075 z^4) ArcTan[z^(1/4)] - 69615 z^(5/4) (13923 - 154700 z + 517650 z^2 - 669900 z^3 + 295075 z^4) ArcTanh[z^(1/4)])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["21", "4"]]], ",", "5", ",", RowBox[List["-", FractionBox["1", "4"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "8388608"], RowBox[List["(", RowBox[List["8388608", "+", RowBox[List["880803840", " ", "z"]], "-", RowBox[List["14327498850", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["57983184350", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["85053518550", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["41083292250", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["69615", " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["13923", "-", RowBox[List["154700", " ", "z"]], "+", RowBox[List["517650", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["669900", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["295075", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], "]"]]]], "-", RowBox[List["69615", " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["13923", "-", RowBox[List["154700", " ", "z"]], "+", RowBox[List["517650", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["669900", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["295075", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["ArcTanh", "[", SuperscriptBox["z", RowBox[List["1", "/", "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> 21 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mn> 5 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </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;21&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;5&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;1&quot;, &quot;4&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> <mn> 8388608 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 41083292250 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 85053518550 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 57983184350 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 14327498850 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 69615 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 295075 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 669900 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 517650 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 154700 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 13923 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 69615 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 295075 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 669900 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 517650 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 154700 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 13923 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 880803840 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 8388608 </mn> </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'> 21 <sep /> 4 </cn> </apply> <cn type='integer'> 5 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8388608 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 41083292250 </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'> 85053518550 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 57983184350 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14327498850 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 69615 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 295075 </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'> 669900 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 517650 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 154700 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 13923 </cn> </apply> <apply> <arctan /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 69615 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 295075 </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'> 669900 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 517650 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 154700 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 13923 </cn> </apply> <apply> <arctanh /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 880803840 </cn> <ci> z </ci> </apply> <cn type='integer'> 8388608 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["21", "4"]]], ",", "5", ",", RowBox[List["-", FractionBox["1", "4"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["8388608", "+", RowBox[List["880803840", " ", "z"]], "-", RowBox[List["14327498850", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["57983184350", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["85053518550", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["41083292250", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["69615", " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["13923", "-", RowBox[List["154700", " ", "z"]], "+", RowBox[List["517650", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["669900", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["295075", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], "]"]]]], "-", RowBox[List["69615", " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["13923", "-", RowBox[List["154700", " ", "z"]], "+", RowBox[List["517650", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["669900", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["295075", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["ArcTanh", "[", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], "]"]]]]]], "8388608"]]]]]


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





2007-05-02