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AiryBiPrime




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.08.06.0024.01









  


  








Input Form





AiryBiPrime[z] \[Proportional] (1/(2 Sqrt[Pi] (-z^3)^(7/12))) (((-(-1)^(11/12)) ((-z^3)^(2/3) + (-1)^(2/3) z^2) (Sum[((Pochhammer[-(1/6), k] Pochhammer[7/6, k])/k!) ((3 I)/(4 Sqrt[-z^3]))^k, {k, 0, n}] + O[1/z^(3 ((n + 1)/2))]))/ E^((2/3) I Sqrt[-z^3]) + (-1)^(1/12) E^((2/3) I Sqrt[-z^3]) ((-z^3)^(2/3) + z^2/(-1)^(2/3)) (Sum[((Pochhammer[-(1/6), k] Pochhammer[7/6, k])/k!) (-((3 I)/(4 Sqrt[-z^3])))^k, {k, 0, n}] + O[1/z^(3 ((n + 1)/2))])) /; (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0








Standard Form





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MathML Form







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/> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryBiPrime", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["11", "/", "12"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "6"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]"]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "2"]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "12"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "/", "3"]]], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "6"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]"]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "2"]]], "]"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["7", "/", "12"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2003-08-21