Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











AiryAi






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryAi[z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/03.05.06.0031.01









  


  










Input Form





AiryAi[z] == Sum[(3^(k - 4/3)/(Subscript[z, 0]^k k!)) (3^(2/3) Gamma[1/3] HypergeometricPFQRegularized[{1/3, 1}, {(1 - k)/3, (2 - k)/3, 1 - k/3}, Subscript[z, 0]^3/9] - Subscript[z, 0] Gamma[2/3] HypergeometricPFQRegularized[{2/3, 1}, {(2 - k)/3, 1 - k/3, (4 - k)/3}, Subscript[z, 0]^3/9]) (z - Subscript[z, 0])^k, {k, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["3", RowBox[List["k", "-", FractionBox["4", "3"]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "k"]], "3"], ",", FractionBox[RowBox[List["2", "-", "k"]], "3"], ",", RowBox[List["1", "-", FractionBox["k", "3"]]]]], "}"]], ",", FractionBox[SubsuperscriptBox["z", "0", "3"], "9"]]], "]"]]]], "-", RowBox[List[SubscriptBox["z", "0"], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "-", "k"]], "3"], ",", RowBox[List["1", "-", FractionBox["k", "3"]]], ",", FractionBox[RowBox[List["4", "-", "k"]], "3"]]], "}"]], ",", FractionBox[SubsuperscriptBox["z", "0", "3"], "9"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> k </mi> <mn> 3 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 3 </mn> </msubsup> <mn> 9 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;3&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;3&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;3&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;3&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;k&quot;, &quot;3&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SubsuperscriptBox[&quot;z&quot;, &quot;0&quot;, &quot;3&quot;], &quot;9&quot;], HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> k </mi> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 3 </mn> </msubsup> <mn> 9 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;3&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;2&quot;, &quot;3&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;3&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;k&quot;, &quot;3&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;4&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;3&quot;], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SubsuperscriptBox[&quot;z&quot;, &quot;0&quot;, &quot;3&quot;], &quot;9&quot;], HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> AiryAi </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='rational'> 1 <sep /> 3 </cn> <cn type='integer'> 1 </cn> </list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 9 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='integer'> 1 </cn> </list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <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> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["k", "-", FractionBox["4", "3"]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "k"]], "3"], ",", FractionBox[RowBox[List["2", "-", "k"]], "3"], ",", RowBox[List["1", "-", FractionBox["k", "3"]]]]], "}"]], ",", FractionBox[SubsuperscriptBox["zz", "0", "3"], "9"]]], "]"]]]], "-", RowBox[List[SubscriptBox["zz", "0"], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "-", "k"]], "3"], ",", RowBox[List["1", "-", FractionBox["k", "3"]]], ",", FractionBox[RowBox[List["4", "-", "k"]], "3"]]], "}"]], ",", FractionBox[SubsuperscriptBox["zz", "0", "3"], "9"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]










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





2007-05-02