html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 AiryAi

 http://functions.wolfram.com/03.05.06.0021.01

 Input Form

 AiryAi[z] \[Proportional] (1/((-z^3)^(5/12) (2 Sqrt[3 Pi]))) (((-1)^(1/12) ((-z^3)^(1/3) - (-1)^(1/3) z) (Sum[((Pochhammer[1/6, k] Pochhammer[5/6, 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]) + (E^((2/3) I Sqrt[-z^3]) ((-z^3)^(1/3) + (-1)^(2/3) z) (Sum[((Pochhammer[1/6, k] Pochhammer[5/6, k])/k!) (-((3 I)/(4 Sqrt[-z^3])))^k, {k, 0, n}] + O[1/z^(3 ((n + 1)/2))]))/ (-1)^12^(-1)) /; (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List[RowBox[List["-", "5"]], "/", "12"]]], RowBox[List["2", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "12"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[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["1", "/", "3"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "3"]]], " ", "z"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "6"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "6"], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]"]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], ")"]], "k"]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["3", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "/", "2"]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[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["1", "/", "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", "/", "3"]]], "z"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "6"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "6"], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]"]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]]]], ")"]], "k"]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["3", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "/", "2"]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 Ai ( z ) ( - z 3 ) - 5 / 12 2 3 π ( - 1 12 1 3 ( - 2 ) - z 3 ( - z 3 3 - - 1 3 z ) ( k = 0 n ( 1 6 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "6"], ")"]], "k"], Pochhammer] ( 5 6 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "6"], ")"]], "k"], Pochhammer] k ! ( 3 4 - z 3 ) k + O ( 1 z 3 ( n + 1 ) 2 ) ) + 1 - 1 12 2 3 - z 3 ( - z 3 3 + ( - 1 ) 2 / 3 z ) ( k = 0 n ( 1 6 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "6"], ")"]], "k"], Pochhammer] ( 5 6 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "6"], ")"]], "k"], Pochhammer] k ! ( - 3 4 - z 3 ) k + O ( 1 z 3 ( n + 1 ) 2 ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional AiryAi z -1 z 3 -5 12 2 3 1 2 -1 -1 1 12 1 3 -2 -1 z 3 1 2 -1 z 3 1 3 -1 -1 1 3 z k 0 n Pochhammer 1 6 k Pochhammer 5 6 k k -1 3 4 -1 z 3 1 2 -1 k O 1 z 3 n 1 2 -1 -1 1 -1 1 12 -1 2 3 -1 z 3 1 2 -1 z 3 1 3 -1 2 3 z k 0 n Pochhammer 1 6 k Pochhammer 5 6 k k -1 -1 3 4 -1 z 3 1 2 -1 k O 1 z 3 n 1 2 -1 -1 Rule z n [/itex]

 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "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["1", "/", "3"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "3"]]], " ", "z"]]]], ")"]], " ", 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[FractionBox["1", "6"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "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[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["1", "/", "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", "/", "3"]]], " ", "z"]]]], ")"]], " ", 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[FractionBox["1", "6"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "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[RowBox[List["3", " ", "\[Pi]"]]]]]], "/;", 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