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AiryBi






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryBi[z] > Integral representations > Contour integral representations





http://functions.wolfram.com/03.06.07.0004.01









  


  










Input Form





AiryBi[z] == (I/3^(1/6)) ContourIntegral[ (Gamma[s] Gamma[s + 1/3])/(Gamma[s + 1/6] Gamma[s + 2/3] Gamma[5/6 - s] Gamma[1/3 - s])/(z/3^(2/3))^(3 s), \[ScriptCapitalL]["s"]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["AiryBi", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[FractionBox["\[ImaginaryI]", RadicalBox["3", "6"]], RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalL]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "3"]]], "]"]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "6"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["2", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "6"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "3"], "-", "s"]], "]"]], " "]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List[RowBox[List["-", "2"]], "/", "3"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "3"]], "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> <mrow> <mi> Bi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> &#8520; </mi> <mroot> <mn> 3 </mn> <mn> 6 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> &#8466; </mi> </msub> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 3 </mn> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <mi> s </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <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> RowBox </ci> <list> <ms> Bi </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> &#8520; </ms> <apply> <ci> RadicalBox </ci> <ms> 3 </ms> <ms> 6 </ms> </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> 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> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> 3 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> / </ms> <ms> 3 </ms> </list> </apply> </apply> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 3 </ms> </list> </apply> <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> 6 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <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> 2 </ms> <ms> 3 </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> 5 </ms> <ms> 6 </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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 3 </ms> </apply> <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["AiryBi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ContourIntegral", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "3"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List[RowBox[List["-", "2"]], "/", "3"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "3"]], " ", "s"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "6"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["2", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "6"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "3"], "-", "s"]], "]"]]]]], ",", RowBox[List["\[ScriptCapitalL]", "[", "\"s\"", "]"]]]], "]"]]]], SuperscriptBox["3", RowBox[List["1", "/", "6"]]]]]]]]










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





2001-10-29





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