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
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] > Specific values > For some numeric parameters and fixed z > For fixed z and a1=1/8, a2=m/8, a3=1





http://functions.wolfram.com/07.27.03.0336.01









  


  










Input Form





HypergeometricPFQ[{1/8, 3/8, 1}, {9/8, 11/8}, -z] == (3/(32 x^3)) (2 (a x^2 - b) ArcTan[1 - x^2, a x] + 2 (b x^2 + a) ArcTan[1 - x^2, b x] + (b x^2 - a) Log[(1 + b x + x^2)/(1 - b x + x^2)] + (a x^2 + b) Log[(1 + a x + x^2)/(1 - a x + x^2)]) /; a == Sqrt[2 - Sqrt[2]] && b == Sqrt[2 + Sqrt[2]] && x == z^(1/8)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "8"], ",", FractionBox["3", "8"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["9", "8"], ",", FractionBox["11", "8"]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["3", RowBox[List["32", SuperscriptBox["x", "3"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["x", "2"]]], "-", "b"]], ")"]], RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", SuperscriptBox["x", "2"]]], ",", RowBox[List["a", " ", "x"]]]], "]"]]]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["x", "2"]]], "+", "a"]], ")"]], RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", SuperscriptBox["x", "2"]]], ",", RowBox[List["b", " ", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["x", "2"]]], "-", "a"]], ")"]], RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", RowBox[List["b", " ", "x"]], "+", SuperscriptBox["x", "2"]]], RowBox[List["1", "-", RowBox[List["b", " ", "x"]], "+", SuperscriptBox["x", "2"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["x", "2"]]], "+", "b"]], ")"]], RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", RowBox[List["a", " ", "x"]], "+", SuperscriptBox["x", "2"]]], RowBox[List["1", "-", RowBox[List["a", " ", "x"]], "+", SuperscriptBox["x", "2"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", SqrtBox[RowBox[List["2", "-", SqrtBox["2"]]]]]], "\[And]", RowBox[List["b", "\[Equal]", SqrtBox[RowBox[List["2", "+", SqrtBox["2"]]]]]], "\[And]", RowBox[List["x", "\[Equal]", SuperscriptBox["z", RowBox[List["1", "/", "8"]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 9 </mn> <mn> 8 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 11 </mn> <mn> 8 </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;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;8&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;8&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;9&quot;, &quot;8&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;11&quot;, &quot;8&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;z&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 3 </mn> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#10869; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> - </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </msqrt> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#10869; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> + </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </msqrt> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &#10869; </mo> <mroot> <mi> z </mi> <mn> 8 </mn> </mroot> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 1 <sep /> 8 </cn> <cn type='rational'> 3 <sep /> 8 </cn> <cn type='integer'> 1 </cn> </list> <list> <cn type='rational'> 9 <sep /> 8 </cn> <cn type='rational'> 11 <sep /> 8 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> x </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> x </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> b </ci> <ci> x </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> x </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> a </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <eq /> <ci> b </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <eq /> <ci> x </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 8 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "8"], ",", FractionBox["3", "8"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["9", "8"], ",", FractionBox["11", "8"]]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["x", "2"]]], "-", "b"]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", SuperscriptBox["x", "2"]]], ",", RowBox[List["a", " ", "x"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["x", "2"]]], "+", "a"]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", SuperscriptBox["x", "2"]]], ",", RowBox[List["b", " ", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["x", "2"]]], "-", "a"]], ")"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", RowBox[List["b", " ", "x"]], "+", SuperscriptBox["x", "2"]]], RowBox[List["1", "-", RowBox[List["b", " ", "x"]], "+", SuperscriptBox["x", "2"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["x", "2"]]], "+", "b"]], ")"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", RowBox[List["a", " ", "x"]], "+", SuperscriptBox["x", "2"]]], RowBox[List["1", "-", RowBox[List["a", " ", "x"]], "+", SuperscriptBox["x", "2"]]]], "]"]]]]]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["x", "3"]]]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", SqrtBox[RowBox[List["2", "-", SqrtBox["2"]]]]]], "&&", RowBox[List["b", "\[Equal]", SqrtBox[RowBox[List["2", "+", SqrtBox["2"]]]]]], "&&", RowBox[List["x", "\[Equal]", SuperscriptBox["z", RowBox[List["1", "/", "8"]]]]]]]]]]]]]










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





2001-10-29