AiryAi[x]^2 + AiryBi[x]^2 == (1/Pi^(3/2)) Integrate[(1/Sqrt[t]) E^(x t - t^3/12), {t, 0, Infinity}]

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["AiryAi", "[", "x", "]"]], "2"], "+", SuperscriptBox[RowBox[List["AiryBi", "[", "x", "]"]], "2"]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox["1", SqrtBox["t"]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["x", " ", "t"]], "-", FractionBox[SuperscriptBox["t", "3"], "12"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Bi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> t </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> - </mo> <mfrac> <msup> <mi> t </mi> <mn> 3 </mn> </msup> <mn> 12 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <power /> <apply> <ci> AiryAi </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> AiryBi </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> t </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> t </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["AiryAi", "[", "x_", "]"]], "2"], "+", SuperscriptBox[RowBox[List["AiryBi", "[", "x_", "]"]], "2"]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["x", " ", "t"]], "-", FractionBox[SuperscriptBox["t", "3"], "12"]]]], SqrtBox["t"]], RowBox[List["\[DifferentialD]", "t"]]]]]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]

2001-10-29