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/03.05.06.0037.01

 Input Form

 AiryAi[z] \[Proportional] ((-1)^(1/4)/((-z^3)^(5/12) (4 Sqrt[3 Pi]))) ((((-(I + Sqrt[3])) z + (-I + Sqrt[3]) (-z^3)^(1/3))/ E^((2/3) I Sqrt[-z^3]) + I E^((2/3) I Sqrt[-z^3]) ((-I + Sqrt[3]) z - (I + Sqrt[3]) (-z^3)^(1/3))) (Sum[((Pochhammer[1/12, k] Pochhammer[5/12, k] Pochhammer[7/12, k] Pochhammer[11/12, k])/(k! Pochhammer[1/2, k])) (9/(4 z^3))^k, {k, 0, n}] + O[1/z^(3 n + 3)]) + (5/(48 Sqrt[-z^3])) ((I ((-I - Sqrt[3]) z + (-I + Sqrt[3]) (-z^3)^(1/3)))/ E^((2/3) I Sqrt[-z^3]) + E^((2/3) I Sqrt[-z^3]) ((-I + Sqrt[3]) z - (I + Sqrt[3]) (-z^3)^(1/3))) (Sum[((Pochhammer[7/12, k] Pochhammer[11/12, k] Pochhammer[13/12, k] Pochhammer[17/12, k])/(k! Pochhammer[3/2, k])) (9/(4 z^3))^k, {k, 0, n}] + O[1/z^(3 n + 3)])) /; (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List[RowBox[List["-", "5"]], "/", "12"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]]]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["11", "12"], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]], ")"]], "k"]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["3", "n"]], "+", "3"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["5", RowBox[List["48", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "-", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]], "+", " ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", " ", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["11", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["13", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["17", "12"], ",", "k"]], "]"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]], ")"]], "k"]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["3", "n"]], "+", "3"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", 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 - 1 4 4 3 π ( ( - 2 3 - z 3 ( ( - + 3 ) - z 3 3 - ( + 3 ) z ) + 2 3 - z 3 ( ( - + 3 ) z - ( + 3 ) - z 3 3 ) ) ( k = 0 n ( 1 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "12"], ")"]], "k"], Pochhammer] ( 5 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "12"], ")"]], "k"], Pochhammer] ( 7 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["7", "12"], ")"]], "k"], Pochhammer] ( 11 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["11", "12"], ")"]], "k"], Pochhammer] k ! ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 9 4 z 3 ) k + O ( 1 z 3 n + 3 ) ) + 5 48 - z 3 ( - 2 3 - z 3 ( ( - + 3 ) - z 3 3 + ( - - 3 ) z ) + 2 3 - z 3 ( ( - + 3 ) z - ( + 3 ) - z 3 3 ) ) ( k = 0 n ( 7 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["7", "12"], ")"]], "k"], Pochhammer] ( 11 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["11", "12"], ")"]], "k"], Pochhammer] ( 13 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["13", "12"], ")"]], "k"], Pochhammer] ( 17 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["17", "12"], ")"]], "k"], Pochhammer] k ! ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( 9 4 z 3 ) k + O ( 1 z 3 n + 3 ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional AiryAi z -1 z 3 -5 12 -1 1 4 4 3 1 2 -1 -1 2 3 -1 -1 z 3 1 2 -1 3 1 2 -1 z 3 1 3 -1 3 1 2 z 2 3 -1 -1 z 3 1 2 -1 3 1 2 z -1 3 1 2 -1 z 3 1 3 k 0 n Pochhammer 1 12 k Pochhammer 5 12 k Pochhammer 7 12 k Pochhammer 11 12 k k Pochhammer 1 2 k -1 9 4 z 3 -1 k O 1 z 3 n 3 -1 5 48 -1 z 3 1 2 -1 -1 2 3 -1 -1 z 3 1 2 -1 3 1 2 -1 z 3 1 3 -1 -1 3 1 2 z 2 3 -1 -1 z 3 1 2 -1 3 1 2 z -1 3 1 2 -1 z 3 1 3 k 0 n Pochhammer 7 12 k Pochhammer 11 12 k Pochhammer 13 12 k Pochhammer 17 12 k k Pochhammer 3 2 k -1 9 4 z 3 -1 k O 1 z 3 n 3 -1 Rule z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List[RowBox[List["-", "5"]], "/", "12"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]]]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]]]], ")"]], " ", 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", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["11", "12"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List[RowBox[List["3", " ", "n"]], "+", "3"]]]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "-", SqrtBox["3"]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]]]], ")"]], " ", 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["7", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["11", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["13", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["17", "12"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List[RowBox[List["3", " ", "n"]], "+", "3"]]]], "]"]]]], ")"]]]], RowBox[List["48", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]]]]]]], ")"]]]], RowBox[List["4", " ", 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)

 2007-05-02