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; }

 AiryAiPrime

 http://functions.wolfram.com/03.07.06.0037.01

 Input Form

 AiryAiPrime[z] \[Proportional] ((-1)^(3/4)/(4 Sqrt[3 Pi] (-z^3)^(7/12))) ((((-I + Sqrt[3]) z^2 + (I + Sqrt[3]) (-z^3)^(2/3))/ E^((2/3) I Sqrt[-z^3]) + I E^((2/3) I Sqrt[-z^3]) ((I + Sqrt[3]) z^2 + (-I + Sqrt[3]) (-z^3)^(2/3))) (Sum[((Pochhammer[-(1/12), k] Pochhammer[5/12, k] Pochhammer[7/12, k] Pochhammer[13/12, k])/(Pochhammer[1/2, k] k!)) (9/(4 z^3))^k, {k, 0, n}] + O[1/z^(3 ((n + 1)/2))]) - (7/(48 Sqrt[-z^3])) ((I ((-I + Sqrt[3]) z^2 + (I + Sqrt[3]) (-z^3)^(2/3)))/ E^((2/3) I Sqrt[-z^3]) + E^((2/3) I Sqrt[-z^3]) ((I + Sqrt[3]) z^2 + (-I + Sqrt[3]) (-z^3)^(2/3))) (Sum[((Pochhammer[5/12, k] Pochhammer[11/12, k] Pochhammer[13/12, k] Pochhammer[19/12, k])/(k! Pochhammer[3/2, k])) (9/(4 z^3))^k, {k, 0, n}] + O[1/z^(3 ((n + 1)/2))])) /; (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAiPrime", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], RowBox[List["4", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["7", "/", "12"]]]]]], 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"]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", 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RowBox[List[FractionBox["7", "12"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["13", "12"], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["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["3", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "/", "2"]]]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["7", 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"]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "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["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "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["19", "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["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 ) ( - 1 ) 3 / 4 4 3 π ( - z 3 ) 7 / 12 ( ( 2 3 - z 3 ( ( - + 3 ) ( - z 3 ) 2 / 3 + ( + 3 ) z 2 ) + - 2 3 - z 3 ( ( + 3 ) ( - z 3 ) 2 / 3 + ( - + 3 ) z 2 ) ) ( k = 0 n ( - 1 12 ) k TagBox[SubscriptBox[RowBox[List["(", 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] ( 13 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["13", "12"], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! ( 9 4 z 3 ) k + O ( 1 z 3 ( n + 1 ) 2 ) ) - 7 48 - z 3 ( 2 3 - z 3 ( ( - + 3 ) ( - z 3 ) 2 / 3 + ( + 3 ) z 2 ) + - 2 3 - z 3 ( ( + 3 ) ( - z 3 ) 2 / 3 + ( - + 3 ) z 2 ) ) ( k = 0 n ( 5 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "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] ( 19 12 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["19", "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 + 1 ) 2 ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional AiryAiPrime z -1 3 4 4 3 1 2 -1 z 3 7 12 -1 2 3 -1 -1 z 3 1 2 -1 3 1 2 -1 z 3 2 3 3 1 2 z 2 -1 2 3 -1 -1 z 3 1 2 3 1 2 -1 z 3 2 3 -1 3 1 2 z 2 k 0 n Pochhammer -1 1 12 k Pochhammer 5 12 k Pochhammer 7 12 k Pochhammer 13 12 k Pochhammer 1 2 k k -1 9 4 z 3 -1 k O 1 z 3 n 1 2 -1 -1 -1 7 48 -1 z 3 1 2 -1 2 3 -1 -1 z 3 1 2 -1 3 1 2 -1 z 3 2 3 3 1 2 z 2 -1 2 3 -1 -1 z 3 1 2 3 1 2 -1 z 3 2 3 -1 3 1 2 z 2 k 0 n Pochhammer 5 12 k Pochhammer 11 12 k Pochhammer 13 12 k Pochhammer 19 12 k k Pochhammer 3 2 k -1 9 4 z 3 -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["AiryAiPrime", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "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"]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", 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"]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["13", "12"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "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"]]], "]"]]]], ")"]]]], "-", FractionBox[RowBox[List["7", " ", 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["-", 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")"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["7", "/", "12"]]]]]], "/;", 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