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AiryAiPrime






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAiPrime[z] > Integral transforms > Hankel transforms





http://functions.wolfram.com/03.07.22.0007.01









  


  










Input Form





HankelTransform[AiryAiPrime[t], {t, \[Nu]}, z] == (2^(-6 - \[Nu]) z^(1/2 + \[Nu]) (-((1/(Pi Gamma[1 + \[Nu]])) (32 3^(1/6 + \[Nu]) Gamma[1/2 + \[Nu]/3] Gamma[7/6 + \[Nu]/3] HypergeometricPFQ[{1/4 + \[Nu]/6, 7/12 + \[Nu]/6, 3/4 + \[Nu]/6, 13/12 + \[Nu]/6}, {1/3, 2/3, 1/3 + \[Nu]/3, 2/3 + \[Nu]/3, 1 + \[Nu]/3}, -(z^6/36)])) + z^2 Gamma[11/6 + \[Nu]/3] ((16 Gamma[7/6 + \[Nu]/3] HypergeometricPFQ[{7/12 + \[Nu]/6, 11/12 + \[Nu]/6, 13/12 + \[Nu]/6, 17/12 + \[Nu]/6}, {2/3, 4/3, 2/3 + \[Nu]/3, 1 + \[Nu]/3, 4/3 + \[Nu]/3}, -(z^6/36)])/ (Gamma[1 + \[Nu]/3] Gamma[(2 + \[Nu])/3] Gamma[(4 + \[Nu])/3]) - (1/(Pi Gamma[3 + \[Nu]])) (3^(17/6 + \[Nu]) z^2 Gamma[5/2 + \[Nu]/3] HypergeometricPFQ[{11/12 + \[Nu]/6, 5/4 + \[Nu]/6, 17/12 + \[Nu]/6, 7/4 + \[Nu]/6}, {4/3, 5/3, 1 + \[Nu]/3, 4/3 + \[Nu]/3, 5/3 + \[Nu]/3}, -(z^6/36)]))))/3^(\[Nu]/3) /; Re[\[Nu]] > -(3/2)










Standard Form





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MathML Form







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</mi> <mn> 6 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 7 </mn> <mn> 12 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 6 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 6 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 13 </mn> <mn> 12 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </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[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;4&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;5&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;6&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;6&quot;], &quot;+&quot;, FractionBox[&quot;7&quot;, &quot;12&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;6&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;6&quot;], &quot;+&quot;, FractionBox[&quot;13&quot;, &quot;12&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;2&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;3&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;3&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;3&quot;], &quot;+&quot;, FractionBox[&quot;2&quot;, &quot;3&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;3&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[SuperscriptBox[&quot;z&quot;, &quot;6&quot;], &quot;36&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#8459; </ci> <apply> <ci> CompoundExpression </ci> <ci> t </ci> <ci> &#957; </ci> </apply> </apply> <apply> <ci> AiryAiPrime </ci> <ci> t </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -6 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#957; 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</ci> <cn type='rational'> 17 <sep /> 6 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 11 <sep /> 12 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#957; 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</ci> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 7 <sep /> 6 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#957; 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</ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </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> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.