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 AiryBiPrime

 http://functions.wolfram.com/03.08.06.0021.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryBiPrime", "[", RowBox[List["-", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], SqrtBox["\[Pi]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[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["13", "12"], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", 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["7", RowBox[List["48", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"]]], "]"]], " ", 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["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", 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["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["2", "\[Pi]"]], "3"]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 Bi ( - z ) z 4 π ( sin ( 2 z 3 / 2 3 + π 4 ) ( 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 + 3 ) ) + 7 48 z 3 / 2 cos ( 2 z 3 / 2 3 + π 4 ) ( 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] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! ( - 9 4 z 3 ) k + O ( 1 z 3 n + 3 ) ) ) /; "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" < 2 π 3 ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition Proportional AiryBiPrime -1 z z 1 4 1 2 -1 2 z 3 2 3 -1 4 -1 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 -1 9 4 z 3 -1 k O 1 z 3 n 3 -1 7 48 z 3 2 -1 2 z 3 2 3 -1 4 -1 k 0 n Pochhammer 5 12 k Pochhammer 11 12 k Pochhammer 13 12 k Pochhammer 19 12 k Pochhammer 3 2 k k -1 -1 9 4 z 3 -1 k O 1 z 3 n 3 -1 z 2 3 -1 Rule z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryBiPrime", "[", RowBox[List["-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"]]], "]"]], " ", 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[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["13", "12"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", 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", ",", RowBox[List[RowBox[List["3", " ", "n"]], "+", "3"]]]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["7", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "3"], "+", FractionBox["\[Pi]", "4"]]], "]"]], " ", 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["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"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["9", RowBox[List["4", " ", SuperscriptBox["z", "3"]]]]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List[RowBox[List["3", " ", "n"]], "+", "3"]]]], "]"]]]], ")"]]]], RowBox[List["48", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]]], ")"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["2", " ", "\[Pi]"]], "3"]]], "&&", 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