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AiryAi




Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAi[z] > Integral transforms > Laplace transforms





http://functions.wolfram.com/03.05.22.0005.01









  


  








Input Form





LaplaceTransform[AiryAi[t], t, z] == (1/(12 Pi)) ((-2 (-1)^(2/3) Sqrt[3] Gamma[2/3] Gamma[1/3, -(z^3/3)] + (3 I + Sqrt[3]) Gamma[1/3] Gamma[2/3, -(z^3/3)])/E^(z^3/3))








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#8466; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mn> 3 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mn> 3 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <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> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mn> 3 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LaplaceTransform </ci> <apply> <ci> AiryAi </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29