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=2, b1>=-23/4 > For fixed z and a1=2, b1=-17/4





http://functions.wolfram.com/07.22.03.6571.01









  


  










Input Form





HypergeometricPFQ[{2}, {-(17/4), -(3/4)}, -z] == (1/29835) (29835 - 18720 z + 49920 z^2 - 4096 z^3 + 256 Sqrt[Pi] z^(7/4) FresnelC[(2 z^(1/4))/Sqrt[Pi]] ((-165 + 240 z + 32 z^2) Cos[2 Sqrt[z]] + 10 Sqrt[z] (-33 + 4 z) Sin[2 Sqrt[z]]) + 256 Sqrt[Pi] z^(7/4) FresnelS[(2 z^(1/4))/Sqrt[Pi]] (-10 Sqrt[z] (-33 + 4 z) Cos[2 Sqrt[z]] + (-165 + 240 z + 32 z^2) Sin[2 Sqrt[z]]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["17", "4"]]], ",", RowBox[List["-", FractionBox["3", "4"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "29835"], RowBox[List["(", RowBox[List["29835", "-", RowBox[List["18720", " ", "z"]], "+", RowBox[List["49920", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["4096", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["256", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["7", "/", "4"]]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SqrtBox["\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "165"]], "+", RowBox[List["240", " ", "z"]], "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List["10", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "33"]], "+", RowBox[List["4", " ", "z"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["256", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["7", "/", "4"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SqrtBox["\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "10"]], " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "33"]], "+", RowBox[List["4", " ", "z"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "165"]], "+", RowBox[List["240", " ", "z"]], "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", 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> <mn> 2 </mn> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 17 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </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[&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[TagBox[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;17&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;3&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> <mn> 29835 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4096 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 49920 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 256 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </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> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 240 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 165 </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> <mn> 10 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 33 </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> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 256 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <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> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 240 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 165 </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> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 33 </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> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 18720 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 29835 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 2 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 17 <sep /> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 29835 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -4096 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 49920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> FresnelC </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> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 240 </cn> <ci> z </ci> </apply> <cn type='integer'> -165 </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 /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> -33 </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> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </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> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 240 </cn> <ci> z </ci> </apply> <cn type='integer'> -165 </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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> -33 </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> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 18720 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 29835 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["17", "4"]]], ",", RowBox[List["-", FractionBox["3", "4"]]]]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["29835", "-", RowBox[List["18720", " ", "z"]], "+", RowBox[List["49920", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["4096", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["256", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["7", "/", "4"]]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SqrtBox["\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "165"]], "+", RowBox[List["240", " ", "z"]], "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List["10", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "33"]], "+", RowBox[List["4", " ", "z"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["256", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["7", "/", "4"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SqrtBox["\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "10"]], " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "33"]], "+", RowBox[List["4", " ", "z"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "165"]], "+", RowBox[List["240", " ", "z"]], "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]]]], "29835"]]]]]










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





2007-05-02