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AiryBi




Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryBi[z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself





http://functions.wolfram.com/03.06.06.0038.01









  


  








Input Form





AiryBi[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (1/(3^(1/6) Gamma[2/3])) Sum[(1/(Pochhammer[2/3, k] k!)) (z^3/9)^k, {k, 0, n}] + ((3^(1/6) z)/Gamma[1/3]) Sum[(1/(Pochhammer[4/3, k] k!)) (z^3/9)^k, {k, 0, n}] == AiryBi[z] - (1/(3^(1/6) Gamma[2/3])) ((z^3/9)^(n + 1)/((n + 1)! Pochhammer[2/3, n + 1])) HypergeometricPFQ[{1}, {n + 2, n + 5/3}, z^3/9] - ((3^(1/6) z)/Gamma[1/3]) ((z^3/9)^(n + 1)/ ((n + 1)! Pochhammer[4/3, n + 1])) HypergeometricPFQ[{1}, {n + 2, n + 7/3}, z^3/9] && Element[n, Integers] && n >= 0








Standard Form





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MathML Form







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type='integer'> 1 </cn> </list> <list> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='rational'> 7 <sep /> 3 </cn> </apply> </list> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 9 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryBi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], "k"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", "z"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], "k"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]], "\[Equal]", RowBox[List[RowBox[List["AiryBi", "[", "z", "]"]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["5", "3"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "3"], "9"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", "z"]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["7", "3"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "3"], "9"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02