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},{b1,b2},z] > Specific values > For rational parameters with larger denominators and fixed z > For fixed z and a1=-9/4, b1`>=-11/2 > For fixed z and a1=-9/4, b1`=11/2





http://functions.wolfram.com/07.22.03.a9od.01









  


  










Input Form





HypergeometricPFQ[{-(9/4)}, {11/2, 11/4}, -z] == (1/(354410496 z^(9/2))) (7 (Sqrt[Pi] z^(11/4) (-23892975 + 33981120 z + 42923520 z^2 + 3981312 z^3 + 65536 z^4) FresnelS[(2 z^(1/4))/Sqrt[Pi]] + 2 (4 Sqrt[z] (-1190700 - 518400 z - 1296000 z^2 + 1712475 z^3 + 2637744 z^4 + 248064 z^5 + 4096 z^6) Cos[2 Sqrt[z]] + (2381400 - 2138400 z + 362880 z^2 + 392175 z^3 + 2464560 z^4 + 244992 z^5 + 4096 z^6) Sin[2 Sqrt[z]])))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["9", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["11", "2"], ",", FractionBox["11", "4"]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["354410496", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]], RowBox[List["(", RowBox[List["7", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["11", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "23892975"]], "+", RowBox[List["33981120", " ", "z"]], "+", RowBox[List["42923520", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["3981312", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["65536", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1190700"]], "-", RowBox[List["518400", " ", "z"]], "-", RowBox[List["1296000", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1712475", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2637744", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["248064", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4096", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2381400", "-", RowBox[List["2138400", " ", "z"]], "+", RowBox[List["362880", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["392175", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2464560", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["244992", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4096", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["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> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 9 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 11 </mn> <mn> 4 </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;1&quot;], SubscriptBox[&quot;F&quot;, &quot;2&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;9&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[TagBox[RowBox[List[TagBox[FractionBox[&quot;11&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;11&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[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> 354410496 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 9 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 65536 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3981312 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 42923520 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 33981120 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 23892975 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 11 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4096 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 248064 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2637744 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1712475 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1296000 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 518400 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1190700 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4096 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 244992 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2464560 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 392175 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 362880 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2138400 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 2381400 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </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'> 9 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 11 <sep /> 2 </cn> <cn type='rational'> 11 <sep /> 4 </cn> </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'> 354410496 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 65536 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3981312 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 42923520 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 33981120 </cn> <ci> z </ci> </apply> <cn type='integer'> -23892975 </cn> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4096 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 248064 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2637744 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1712475 </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'> 1296000 </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'> 518400 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1190700 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4096 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 244992 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2464560 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 392175 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 362880 </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'> 2138400 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2381400 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </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["9", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["11", "2"], ",", FractionBox["11", "4"]]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["7", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["11", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "23892975"]], "+", RowBox[List["33981120", " ", "z"]], "+", RowBox[List["42923520", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["3981312", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["65536", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1190700"]], "-", RowBox[List["518400", " ", "z"]], "-", RowBox[List["1296000", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1712475", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2637744", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["248064", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4096", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2381400", "-", RowBox[List["2138400", " ", "z"]], "+", RowBox[List["362880", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["392175", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2464560", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["244992", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4096", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["354410496", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]]]]]]










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





2007-05-02