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AiryAi






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAi[z] > Identities > Identities involving determinants





http://functions.wolfram.com/03.05.17.0002.01









  


  










Input Form





D[w[z], {z, 2}] == 2 w[z]^3 + 2 z w[z] + (2 n + 1) /; w[z] == D[Log[Subscript[\[Tau], n + 1][z]/Subscript[\[Tau], n][z]], z] && Subscript[\[Tau], n][z] == Det[Table[D[AiryAi[z], {z, i + j}], {i, 0, n - 1}, {j, 0, n - 1}]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["w", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["w", "[", "z", "]"]], "3"]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["w", "[", "z", "]"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["Log", "[", FractionBox[RowBox[List[SubscriptBox["\[Tau]", RowBox[List["n", "+", "1"]]], "[", "z", "]"]], RowBox[List[SubscriptBox["\[Tau]", "n"], "[", "z", "]"]]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[Tau]", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List["Det", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", RowBox[List["i", "+", "j"]]]], "}"]]], RowBox[List["AiryAi", "[", "z", "]"]]]], ",", RowBox[List["{", RowBox[List["i", ",", "0", ",", RowBox[List["n", "-", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", RowBox[List["n", "-", "1"]]]], "}"]]]], "]"]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> &#964; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msub> <mi> &#964; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> &#964; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <semantics> <mrow> <mo> &#10072; </mo> <msub> <mrow> <mo> ( </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> </mrow> </msup> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> </mrow> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> l </mi> <mo> &#8804; </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </msub> <mo> &#10072; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftBracketingBar]&quot;, SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[RowBox[List[SuperscriptBox[&quot;\[PartialD]&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;l&quot;]]], RowBox[List[&quot;Ai&quot;, &quot;(&quot;, &quot;z&quot;, &quot;)&quot;]]]], RowBox[List[&quot;\[PartialD]&quot;, SuperscriptBox[&quot;z&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;l&quot;]]]]], Rule[MultilineFunction, None]], &quot;)&quot;]], GridBox[List[List[RowBox[List[&quot;0&quot;, &quot;\[LessEqual]&quot;, &quot;k&quot;, &quot;\[LessEqual]&quot;, RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;1&quot;]]]]], List[RowBox[List[&quot;0&quot;, &quot;\[LessEqual]&quot;, &quot;l&quot;, &quot;\[LessEqual]&quot;, RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;1&quot;]]]]]]]], &quot;\[RightBracketingBar]&quot;]], List[Det]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> w </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ln /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#964; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> &#964; </ci> <ci> n </ci> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#964; </ci> <ci> n </ci> </apply> <ci> z </ci> </apply> <apply> <determinant /> <apply> <ci> Subscript </ci> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <apply> <plus /> <ci> k </ci> <ci> l </ci> </apply> </degree> </bvar> <apply> <ci> AiryAi </ci> <ci> z </ci> </apply> </apply> <list> <list> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> k </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> l </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </list> </list> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "2"]], "}"]]]]], RowBox[List["w", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["w", "[", "z", "]"]], "3"]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["w", "[", "z", "]"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z"]]], RowBox[List["Log", "[", FractionBox[RowBox[List[SubscriptBox["\[Tau]", RowBox[List["n", "+", "1"]]], "[", "z", "]"]], RowBox[List[SubscriptBox["\[Tau]", "n"], "[", "z", "]"]]], "]"]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[Tau]", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List["Det", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", RowBox[List["i", "+", "j"]]]], "}"]]]]], RowBox[List["AiryAi", "[", "z", "]"]]]], ",", RowBox[List["{", RowBox[List["i", ",", "0", ",", RowBox[List["n", "-", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", RowBox[List["n", "-", "1"]]]], "}"]]]], "]"]], "]"]]]]]]]]]]]]










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





2001-10-29





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