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AiryAi




Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAi[z] > Integration > Definite integration > Involving the direct function and derivatives





http://functions.wolfram.com/03.05.21.0083.01









  


  








Input Form





Integrate[(Derivative[n][AiryAi][x/\[Tau]^(1/3)] Derivative[m][AiryAi][ y/(t - \[Tau])^(1/3)])/(\[Tau]^((n + 1)/3) (t - \[Tau])^((m + 1)/3)), {\[Tau], 0, t}] == Derivative[n + m - 2][AiryAi][(x + y)/t^(1/3)]/ t^((1/3) (n + m - 1)) /; Element[n, Integers] && Element[m, Integers]








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> <mtext> </mtext> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> t </mi> </msubsup> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> Ai </mi> <semantics> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;n&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mfrac> <mi> x </mi> <mroot> <mi> &#964; </mi> <mn> 3 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> Ai </mi> <semantics> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;m&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mfrac> <mi> y </mi> <mroot> <mrow> <mi> t </mi> <mo> - </mo> <mi> &#964; </mi> </mrow> <mn> 3 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> &#964; </mi> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mi> &#964; </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </mfrac> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#964; </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> t </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> Ai </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;n&quot;, &quot;-&quot;, &quot;2&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mroot> <mi> t </mi> <mn> 3 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> &#964; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> t </ci> </uplimit> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> AiryAi </ci> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <ci> &#964; </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <ci> &#964; </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> </list> </apply> <apply> <ci> D </ci> <apply> <ci> AiryAi </ci> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#964; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#964; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </list> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> &#964; </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#964; </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> AiryAi </ci> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </list> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8484; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8484; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "t_"], RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox["AiryAi", TagBox[RowBox[List["(", "n_", ")"]], Derivative], Rule[MultilineFunction, None]], "[", FractionBox["x_", SuperscriptBox["\[Tau]_", RowBox[List["1", "/", "3"]]]], "]"]], " ", RowBox[List[SuperscriptBox["AiryAi", TagBox[RowBox[List["(", "m_", ")"]], Derivative], Rule[MultilineFunction, None]], "[", FractionBox["y_", SuperscriptBox[RowBox[List["(", RowBox[List["t_", "-", "\[Tau]_"]], ")"]], RowBox[List["1", "/", "3"]]]], "]"]]]], RowBox[List[SuperscriptBox["\[Tau]_", FractionBox[RowBox[List["n_", "+", "1"]], "3"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["t_", "-", "\[Tau]_"]], ")"]], FractionBox[RowBox[List["m_", "+", "1"]], "3"]]]]], RowBox[List["\[DifferentialD]", "\[Tau]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["n", "+", "m", "-", "1"]], ")"]]]]], " ", RowBox[List[SuperscriptBox["AiryAi", TagBox[RowBox[List["(", RowBox[List["n", "+", "m", "-", "2"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", FractionBox[RowBox[List["x", "+", "y"]], SuperscriptBox["t", RowBox[List["1", "/", "3"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]]]


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





2002-12-18