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 AiryBiPrime

 http://functions.wolfram.com/03.08.06.0054.01

 Input Form

 AiryBiPrime[z] \[Proportional] Piecewise[{{(((-1)^(1/4) z^(1/4))/Sqrt[2 Pi]) (Cosh[(2 z^(3/2))/3] - I Sinh[(2 z^(3/2))/3]), Arg[z] <= -((2 Pi)/3)}, {((I z^(1/4))/(2 Sqrt[Pi])) ((1 - 2 I) Cosh[(2 z^(3/2))/3] - (1 + 2 I) Sinh[(2 z^(3/2))/3]), Inequality[-((2 Pi)/3), Less, Arg[z], LessEqual, 0]}, {(-((I z^(1/4))/(2 Sqrt[Pi]))) ((1 + 2 I) Cosh[(2 z^(3/2))/3] - (1 - 2 I) Sinh[(2 z^(3/2))/3]), Inequality[0, Less, Arg[z], LessEqual, (2 Pi)/3]}}, (-(((-1)^(3/4) z^(1/4))/Sqrt[2 Pi])) (Cosh[(2 z^(3/2))/3] + I Sinh[(2 z^(3/2))/3])] /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryBiPrime", "[", "z", "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], " "]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], ",", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List["-", FractionBox[RowBox[List["2", "\[Pi]"]], "3"]]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", "\[Pi]"]], "3"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], ",", RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox[RowBox[List["2", "\[Pi]"]], "3"]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], " "]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 Bi ( z ) - 1 4 z 4 2 π ( cosh ( 2 z 3 / 2 3 ) - sinh ( 2 z 3 / 2 3 ) ) arg ( z ) - 2 π 3 z 4 2 π ( ( 1 - 2 ) cosh ( 2 z 3 / 2 3 ) - ( 1 + 2 ) sinh ( 2 z 3 / 2 3 ) ) - 2 π 3 < arg ( z ) 0 - z 4 2 π ( ( 1 + 2 ) cosh ( 2 z 3 / 2 3 ) - ( 1 - 2 ) sinh ( 2 z 3 / 2 3 ) ) 0 < arg ( z ) 2 π 3 - ( - 1 ) 3 / 4 z 4 2 π ( cosh ( 2 z 3 / 2 3 ) + sinh ( 2 z 3 / 2 3 ) ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional AiryBiPrime z -1 1 4 z 1 4 2 1 2 -1 2 z 3 2 3 -1 -1 2 z 3 2 3 -1 z -1 2 3 -1 z 1 4 2 1 2 -1 1 -2 2 z 3 2 3 -1 -1 1 2 2 z 3 2 3 -1 Inequality -1 2 3 -1 z 0 -1 z 1 4 2 1 2 -1 1 2 2 z 3 2 3 -1 -1 1 -2 2 z 3 2 3 -1 Inequality 0 z 2 3 -1 -1 -1 3 4 z 1 4 2 1 2 -1 2 z 3 2 3 -1 2 z 3 2 3 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryBiPrime", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]]]]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]], List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]], RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox[RowBox[List["2", " ", "\[Pi]"]], "3"]]]], List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02