AiryAi[z] \[Proportional] (1/(3^(2/3) Gamma[2/3])) (1 + z^3/6 + z^6/180 + \[Ellipsis]) - (z/(3^(1/3) Gamma[1/3])) (1 + z^3/12 + z^6/504 + \[Ellipsis]) /; (z -> 0)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", "3"], "6"], "+", FractionBox[SuperscriptBox["z", "6"], "180"], "+", "\[Ellipsis]"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["z", " "]], RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", "3"], "12"], "+", FractionBox[SuperscriptBox["z", "6"], "504"], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mn> 6 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mn> 180 </mn> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <mroot> <mn> 3 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mn> 12 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mn> 504 </mn> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> AiryAi </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 180 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 504 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["z", "3"], "6"], "+", FractionBox[SuperscriptBox["z", "6"], "180"], "+", "\[Ellipsis]"]], RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]]], "-", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", "3"], "12"], "+", FractionBox[SuperscriptBox["z", "6"], "504"], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]

2001-10-29