Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











AiryBiPrime






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryBiPrime[z] > Series representations > Asymptotic series expansions > Expansions inside Stokes sectors > In trigonometric form ||| In trigonometric form





http://functions.wolfram.com/03.08.06.0012.01









  


  










Input Form





AiryBiPrime[z/E^((Pi I)/3)] \[Proportional] E^((I Pi)/6) Sqrt[2/Pi] z^(1/4) (Cos[(2 z^(3/2))/3 + Pi/4 + (I Log[2])/2] (1 + O[1/z^3]) - (7/(48 z^(3/2))) Sin[(2 z^(3/2))/3 + Pi/4 + (I Log[2])/2] (1 + O[1/z^3])) /; Abs[Arg[z]] < (2 Pi)/3 && (Abs[z] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryBiPrime", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "3"]]]], " ", "z"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "6"]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "2", "]"]]]], "2"]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "3"]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["7", RowBox[List["48", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "2", "]"]]]], "2"]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "3"]], "]"]]]], ")"]]]]]], " ", ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["2", "\[Pi]"]], "3"]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










MathML Form







<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> Bi </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mtext> </mtext> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 6 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mn> 3 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 7 </mn> <mtext> </mtext> </mrow> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mn> 3 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> AiryBiPrime </ci> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <arg /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryBiPrime", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]], " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "6"]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", "2", "]"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], ")"]]]], "-", FractionBox[RowBox[List["7", " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", "2", "]"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], ")"]]]], RowBox[List["48", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["2", " ", "\[Pi]"]], "3"]]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]










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





2001-10-29