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 AiryBiPrime

 http://functions.wolfram.com/03.08.06.0049.01

 Input Form

 AiryBiPrime[z] \[Proportional] (1/(2 Sqrt[2 Pi] z (-z^3)^(5/12))) ((Sqrt[-z^3] ((1 + Sqrt[3]) z + (-1 + Sqrt[3]) (-z^3)^(1/3)) Cosh[(2 z^(3/2))/3] + z^(3/2) ((-1 + Sqrt[3]) z + (1 + Sqrt[3]) (-z^3)^(1/3)) Sinh[(2 z^(3/2))/3]) (1 - 455/(4608 z^3) - 40415375/(127401984 z^6) - 6183948445675/(1761205026816 z^9) + O[1/z^12]) - (7/(48 z^(3/2))) (z^(3/2) ((-1 + Sqrt[3]) z + (1 + Sqrt[3]) (-z^3)^(1/3)) Cosh[(2 z^(3/2))/3] + Sqrt[-z^3] ((1 + Sqrt[3]) z + (-1 + Sqrt[3]) (-z^3)^(1/3)) Sinh[(2 z^(3/2))/3]) (1 + 13585/(13824 z^3) + 823318925/(127401984 z^6) + 189935559402875/(1761205026816 z^9) + O[1/z^12])) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryBiPrime", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["5", "/", "12"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", FractionBox["455", RowBox[List["4608", " ", SuperscriptBox["z", "3"]]]], "-", FractionBox["40415375", RowBox[List["127401984", " ", SuperscriptBox["z", "6"]]]], "-", FractionBox["6183948445675", RowBox[List["1761205026816", " ", SuperscriptBox["z", "9"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "12"]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["7", RowBox[List["48", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", FractionBox["13585", RowBox[List["13824", " ", SuperscriptBox["z", "3"]]]], "+", FractionBox["823318925", RowBox[List["127401984", " ", SuperscriptBox["z", "6"]]]], "+", FractionBox["189935559402875", RowBox[List["1761205026816", " ", SuperscriptBox["z", "9"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "12"]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 Bi ( z ) 1 2 2 π z ( - z 3 ) 5 / 12 ( ( z 3 / 2 ( ( 1 + 3 ) - z 3 3 + ( - 1 + 3 ) z ) sinh ( 2 z 3 / 2 3 ) + - z 3 ( ( - 1 + 3 ) - z 3 3 + ( 1 + 3 ) z ) cosh ( 2 z 3 / 2 3 ) ) ( 1 - 455 4608 z 3 - 40415375 127401984 z 6 - 6183948445675 1761205026816 z 9 + O ( 1 z 12 ) ) - 7 48 z 3 / 2 ( z 3 / 2 ( ( 1 + 3 ) - z 3 3 + ( - 1 + 3 ) z ) cosh ( 2 z 3 / 2 3 ) + - z 3 ( ( - 1 + 3 ) - z 3 3 + ( 1 + 3 ) z ) sinh ( 2 z 3 / 2 3 ) ) ( 1 + 13585 13824 z 3 + 823318925 127401984 z 6 + 189935559402875 1761205026816 z 9 + O ( 1 z 12 ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional AiryBiPrime z 1 2 2 1 2 z -1 z 3 5 12 -1 z 3 2 1 3 1 2 -1 z 3 1 3 -1 3 1 2 z 2 z 3 2 3 -1 -1 z 3 1 2 -1 3 1 2 -1 z 3 1 3 1 3 1 2 z 2 z 3 2 3 -1 1 -1 455 4608 z 3 -1 -1 40415375 127401984 z 6 -1 -1 6183948445675 1761205026816 z 9 -1 O 1 z 12 -1 -1 7 48 z 3 2 -1 z 3 2 1 3 1 2 -1 z 3 1 3 -1 3 1 2 z 2 z 3 2 3 -1 -1 z 3 1 2 -1 3 1 2 -1 z 3 1 3 1 3 1 2 z 2 z 3 2 3 -1 1 13585 13824 z 3 -1 823318925 127401984 z 6 -1 189935559402875 1761205026816 z 9 -1 O 1 z 12 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryBiPrime", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox["455", RowBox[List["4608", " ", SuperscriptBox["z", "3"]]]], "-", FractionBox["40415375", RowBox[List["127401984", " ", SuperscriptBox["z", "6"]]]], "-", FractionBox["6183948445675", RowBox[List["1761205026816", " ", SuperscriptBox["z", "9"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "12"]], "]"]]]], ")"]]]], "-", FractionBox[RowBox[List["7", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["13585", RowBox[List["13824", " ", SuperscriptBox["z", "3"]]]], "+", FractionBox["823318925", RowBox[List["127401984", " ", SuperscriptBox["z", "6"]]]], "+", FractionBox["189935559402875", RowBox[List["1761205026816", " ", SuperscriptBox["z", "9"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "12"]], "]"]]]], ")"]]]], RowBox[List["48", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["5", "/", "12"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02