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
Hypergeometric0F1Regularized




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric0F1Regularized[b,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/07.18.07.0010.01









  


  








Input Form





Hypergeometric0F1Regularized[b, z] == (1/(2 I)) ContourIntegral[Gamma[s]/(Gamma[1/2 + s] Gamma[1/2 - s] Gamma[b - s])/z^s, \[ScriptCapitalL]["s"]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", "\[ImaginaryI]"]]], RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalL]"], RowBox[List[FractionBox[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "-", "s"]], "]"]]]]], SuperscriptBox["z", RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]]]]








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> 0 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mo> &#8202; </mo> <mo> ; </mo> <mi> b </mi> <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;0&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;b&quot;, Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric0F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> &#8466; </mi> </msub> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <apply> <ci> FormBox </ci> <ms> 0 </ms> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> FormBox </ci> <ms> 1 </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> &#62368; </ms> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> b </ms> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ci> Hypergeometric0F1Regularized </ci> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#8520; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8747; </ms> </apply> <ms> &#8466; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <ms> s </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> s </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> b </ms> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> s </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["ContourIntegral", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox["z", RowBox[List["-", "s"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "s"]], "]"]]]]], ",", RowBox[List["\[ScriptCapitalL]", "[", "\"s\"", "]"]]]], "]"]], RowBox[List["2", " ", "\[ImaginaryI]"]]]]]]]


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





2001-10-29