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AiryAi




Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAi[z] > Series representations > Asymptotic series expansions > Expansions for any z in trigonometric form > Using trigonometric functions with branch cut-containing arguments





http://functions.wolfram.com/03.05.06.0026.01









  


  








Input Form





AiryAi[z] \[Proportional] (1/((-z^3)^(5/12) (2 Sqrt[3 Pi]))) ((Sqrt[3] ((-z^3)^(1/3) - z) Cos[(2 Sqrt[-z^3])/3 - Pi/4] + ((-z^3)^(1/3) + z) Cos[(2 Sqrt[-z^3])/3 + Pi/4]) HypergeometricPFQ[{1/12, 5/12, 7/12, 11/12}, {1/2}, 9/(4 z^3)] + (5/(48 Sqrt[-z^3])) ((z + (-z^3)^(1/3)) Cos[(2 Sqrt[-z^3])/3 - Pi/4] + Sqrt[3] (z - (-z^3)^(1/3)) Cos[(2 Sqrt[-z^3])/3 + Pi/4]) HypergeometricPFQ[{7/12, 11/12, 13/12, 17/12}, {3/2}, 9/(4 z^3)]) /; (Abs[z] -> Infinity)








Standard Form





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MathML Form







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type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 7 <sep /> 12 </cn> <cn type='rational'> 11 <sep /> 12 </cn> <cn type='rational'> 13 <sep /> 12 </cn> <cn type='rational'> 17 <sep /> 12 </cn> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> </list> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List[RowBox[List["-", "5"]], "/", "12"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]], "-", "z"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]], "3"], "-", FractionBox["\[Pi]", "4"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]], "+", "z"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]], "3"], "+", FractionBox["\[Pi]", "4"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "12"], ",", FractionBox["5", "12"], ",", FractionBox["7", "12"], ",", FractionBox["11", "12"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]]]], "]"]]]], "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["z", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]], "3"], "-", FractionBox["\[Pi]", "4"]]], "]"]]]], "+", RowBox[List[SqrtBox["3"], " ", RowBox[List["(", RowBox[List["z", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]], "3"], "+", FractionBox["\[Pi]", "4"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "12"], ",", FractionBox["11", "12"], ",", FractionBox["13", "12"], ",", FractionBox["17", "12"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]]]], "]"]]]], RowBox[List["48", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2003-08-21