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 > Specialized values > For fixed b1, z





http://functions.wolfram.com/07.22.03.0024.01









  


  








Input Form





HypergeometricPFQ[{3/2}, {b, 1 - b}, z] == (1/2) Pi Sqrt[z] Csc[Pi b] (Sqrt[z] BesselI[-1 - b, Sqrt[z]] BesselI[-1 + b, Sqrt[z]] + 2 Sqrt[z] BesselI[-b, Sqrt[z]] BesselI[b, Sqrt[z]] + Sqrt[z] BesselI[1 - b, Sqrt[z]] BesselI[1 + b, Sqrt[z]] + BesselI[-1 + b, Sqrt[z]] BesselI[-b, Sqrt[z]] (1 + 2 b) + BesselI[1 - b, Sqrt[z]] BesselI[b, Sqrt[z]] (1 + 2 b))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", SqrtBox["z"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["b", ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "+", "b"]], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["b", ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]]]], ")"]]]]]]]]








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> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;b&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&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> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mi> b </mi> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mi> b </mi> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 3 <sep /> 2 </cn> </list> <list> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <csc /> <apply> <times /> <pi /> <ci> b </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </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["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", RowBox[List["1", "-", "b_"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", SqrtBox["z"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["b", ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "+", "b"]], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["b", ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]]]], ")"]]]]]]]]


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





2001-10-29