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AiryAi






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAi[z] > Zeros





http://functions.wolfram.com/03.05.30.0001.01









  


  










Input Form





AiryAi[z] == 0 /; z == Subscript[z, k] && Element[k, Integers] && k >= 0 && Subscript[z, k] == f[3 Pi ((4 k - 1)/8)] && f[d] == (-d^(2/3)) (1 + 5/(48 d^2) - 5/(36 d^4) + 77125/(82944 d^6) - 108056875/(6967296 d^8) + 162375596875/(334430208 d^10) - 1622671914671875/(66217181184 d^12) + 150126478779573265625/ (82639042117632 d^14) - 644932726927939889453125/ (3470839768940544 d^16) + 13042116997445589075044921875/ (520200964553048064 d^18) - 569789860268573944980176052734375/ (132083753999696658432 d^20))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["z", "\[Equal]", SubscriptBox["z", "k"]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["z", "k"], "\[Equal]", RowBox[List["f", "[", RowBox[List["3", "\[Pi]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", "k"]], "-", "1"]], ")"]], "/", "8"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["f", "[", "d", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["d", "^", RowBox[List["(", RowBox[List["2", "/", "3"]], ")"]]]]]], "*", RowBox[List["(", RowBox[List["1", "+", RowBox[List["5", "/", RowBox[List["(", RowBox[List["48", "*", RowBox[List["d", "^", "2"]]]], ")"]]]], "-", RowBox[List["5", "/", RowBox[List["(", RowBox[List["36", "*", RowBox[List["d", "^", "4"]]]], ")"]]]], "+", RowBox[List["77125", "/", RowBox[List["(", RowBox[List["82944", "*", RowBox[List["d", "^", "6"]]]], ")"]]]], "-", RowBox[List["108056875", "/", RowBox[List["(", RowBox[List["6967296", "*", RowBox[List["d", "^", "8"]]]], ")"]]]], "+", RowBox[List["162375596875", "/", RowBox[List["(", RowBox[List["334430208", "*", RowBox[List["d", "^", "10"]]]], ")"]]]], "-", RowBox[List["1622671914671875", "/", RowBox[List["(", RowBox[List["66217181184", "*", RowBox[List["d", "^", "12"]]]], ")"]]]], "+", RowBox[List["150126478779573265625", "/", RowBox[List["(", RowBox[List["82639042117632", "*", RowBox[List["d", "^", "14"]]]], ")"]]]], "-", RowBox[List["644932726927939889453125", "/", RowBox[List["(", RowBox[List["3470839768940544", "*", RowBox[List["d", "^", "16"]]]], ")"]]]], "+", RowBox[List["13042116997445589075044921875", "/", RowBox[List["(", RowBox[List["520200964553048064", "*", RowBox[List["d", "^", "18"]]]], ")"]]]], "-", RowBox[List["569789860268573944980176052734375", "/", RowBox[List["(", RowBox[List["132083753999696658432", "*", RowBox[List["d", "^", "20"]]]], ")"]]]]]], ")"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#10869; </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 5 </mn> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 5 </mn> <mrow> <mn> 36 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 77125 </mn> <mrow> <mn> 82944 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 6 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 108056875 </mn> <mrow> <mn> 6967296 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 8 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 162375596875 </mn> <mrow> <mn> 334430208 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 10 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1622671914671875 </mn> <mrow> <mn> 66217181184 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 12 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 150126478779573265625 </mn> <mrow> <mn> 82639042117632 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 14 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 644932726927939889453125 </mn> <mrow> <mn> 3470839768940544 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 16 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 13042116997445589075044921875 </mn> <mrow> <mn> 520200964553048064 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 18 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 569789860268573944980176052734375 </mn> <mrow> <mn> 132083753999696658432 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 20 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> AiryAi </ci> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> <apply> <in /> <ci> k </ci> <ci> &#8469; </ci> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> <apply> <ci> f </ci> <apply> <times /> <cn type='rational'> 3 <sep /> 8 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> f </ci> <ci> d </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> d </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 77125 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 82944 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 108056875 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6967296 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 162375596875 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 334430208 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1622671914671875 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 66217181184 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 12 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 150126478779573265625 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 82639042117632 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 14 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 644932726927939889453125 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3470839768940544 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 16 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 13042116997445589075044921875 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 520200964553048064 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 18 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 569789860268573944980176052734375 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 132083753999696658432 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 20 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["z", "\[Equal]", SubscriptBox["z", "k"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["z", "k"], "\[Equal]", RowBox[List["f", "[", RowBox[List[FractionBox["3", "8"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "-", "1"]], ")"]]]], "]"]]]], "&&", RowBox[List[RowBox[List["f", "[", "d", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["d", RowBox[List["2", "/", "3"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["5", RowBox[List["48", " ", SuperscriptBox["d", "2"]]]], "-", FractionBox["5", RowBox[List["36", " ", SuperscriptBox["d", "4"]]]], "+", FractionBox["77125", RowBox[List["82944", " ", SuperscriptBox["d", "6"]]]], "-", FractionBox["108056875", RowBox[List["6967296", " ", SuperscriptBox["d", "8"]]]], "+", FractionBox["162375596875", RowBox[List["334430208", " ", SuperscriptBox["d", "10"]]]], "-", FractionBox["1622671914671875", RowBox[List["66217181184", " ", SuperscriptBox["d", "12"]]]], "+", FractionBox["150126478779573265625", RowBox[List["82639042117632", " ", SuperscriptBox["d", "14"]]]], "-", FractionBox["644932726927939889453125", RowBox[List["3470839768940544", " ", SuperscriptBox["d", "16"]]]], "+", FractionBox["13042116997445589075044921875", RowBox[List["520200964553048064", " ", SuperscriptBox["d", "18"]]]], "-", FractionBox["569789860268573944980176052734375", RowBox[List["132083753999696658432", " ", SuperscriptBox["d", "20"]]]]]], ")"]]]]]]]]]]]]]]










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





2001-10-29





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