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/03.05.06.0048.01

 Input Form

 AiryAi[z] \[Proportional] (1/(4 Sqrt[3 Pi])) ((Sqrt[2]/((-z^3)^(5/12) 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]) 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, Infinity}] - (5/(24 Sqrt[2] (-z^3)^(17/12))) (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]) 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, Infinity}]) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List[RowBox[List["-", "5"]], "/", "12"]]]]], 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["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["(", 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["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", ",", "\[Infinity]"]], "}"]]]], "]"]]]], "-", RowBox[List[FractionBox["5", RowBox[List["24", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["17", "/", "12"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]], " ", RowBox[List["(", 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["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["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", ",", "\[Infinity]"]], "}"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 Ai ( z ) 1 4 3 π ( 2 ( - z 3 ) - 5 / 12 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 ) ) k = 0 ( 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 - 5 24 2 ( - z 3 ) 17 / 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 ) ) k = 0 ( 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 ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional AiryAi z 1 4 3 1 2 -1 2 1 2 -1 z 3 -5 12 z 3 2 -1 z 3 2 1 3 1 2 -1 z 3 1 3 1 -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 1 3 1 2 z 2 z 3 2 3 -1 k 0 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 -1 5 24 2 1 2 -1 z 3 17 12 -1 z 3 2 1 3 1 2 -1 z 3 1 3 1 -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 1 3 1 2 z 2 z 3 2 3 -1 k 0 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 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List[RowBox[List["-", "5"]], "/", "12"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", 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["(", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], 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"]], "]"]]]]]]]]], SuperscriptBox["z", RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "3"]]]], " ", RowBox[List["(", 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["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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], 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["24", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["17", "/", "12"]]]]]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02