AiryAiPrime[c (d z^n)^m] == (1/2) ((d z^3)^(2 m)/(d^(2 m) z^(6 m)) + 1) AiryAiPrime[c d^m z^(3 m)] - ((1 - (d z^3)^(2 m)/(d^(2 m) z^(6 m))) AiryBiPrime[c d^m z^(3 m)])/(2 Sqrt[3]) /; IntegerQ[3 m]

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<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> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mrow> <msup> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> Ai </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mrow> <msup> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> Bi </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> IntegerQ </mi> <mo> [ </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ] </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> AiryAiPrime </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 6 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> AiryAiPrime </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 6 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> AiryBiPrime </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> IntegerQ </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29