Derivative of the Airy function Ai: Integral transforms (formula 03.07.22.0004)

AiryAiPrime

$Mathematica Notation$

Bessel-Type Functions

AiryAiPrime[z]

Integral transforms

Fourier sin transforms

http://functions.wolfram.com/03.07.22.0004.01

Input Form

FourierSinTransform[AiryAiPrime[t], t, z] == (-(1/(12 Sqrt[2] Pi^(3/2)))) (8 Pi z Cos[z^3/3] - 2 3^(5/6) z^3 Gamma[1/3] HypergeometricPFQ[{1}, {5/6, 4/3}, -(z^6/36)] + 3 3^(1/6) z^5 Gamma[2/3] HypergeometricPFQ[{1}, {7/6, 5/3}, -(z^6/36)])

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> ℱ𝓈 </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <msup> <mi> Ai </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mroot> <mn> 3 </mn> <mn> 6 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mn> 36 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "6"], "36"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mn> 3 </mn> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 6 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mn> 36 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["4", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "6"], "36"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> ℱ𝓈 </ci> <ci> t </ci> </apply> <apply> <ci> AiryAiPrime </ci> <ci> t </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <cn type='rational'> 7 <sep /> 6 </cn> <cn type='rational'> 5 <sep /> 3 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 36 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 5 <sep /> 6 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <cn type='rational'> 5 <sep /> 6 </cn> <cn type='rational'> 4 <sep /> 3 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 36 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <pi /> <apply> <cos /> <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> <ci> z </ci> </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