AiryAiPrime[z] == (-(3^(1/6)/(2 Pi))) (1/(2 Pi I)) Integrate[(Gamma[s] Gamma[s + 2/3])/(z/3^(2/3))^(3 s), {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; 0 < \[Gamma]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAiPrime", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], RowBox[List["2", " ", "\[Pi]"]]]]], FractionBox["1", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["2", "3"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List[RowBox[List["-", "2"]], "/", "3"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "3"]], "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]], "/;", " ", RowBox[List["0", "<", "\[Gamma]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> Ai </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mroot> <mn> 3 </mn> <mn> 6 </mn> </mroot> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mrow> <mi> &#947; </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </msubsup> <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> 2 </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> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> &#947; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> AiryAiPrime </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> s </ci> </bvar> <lowlimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> -2 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -3 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> &#947; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAiPrime", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["2", "3"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List[RowBox[List["-", "2"]], "/", "3"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "3"]], " ", "s"]]]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]], "/;", RowBox[List["0", "<", "\[Gamma]"]]]]]]]]

2001-10-29