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] > Integration > Indefinite integration > Involving only one direct function





http://functions.wolfram.com/03.05.21.0001.01









  


  








Input Form





Integrate[AiryAi[a z], z] == (z Gamma[1/3] HypergeometricPFQ[{1/3}, {2/3, 4/3}, (a^3 z^3)/9])/ (3 3^(2/3) Gamma[2/3] Gamma[4/3]) - (a z^2 Gamma[2/3] HypergeometricPFQ[{2/3}, {4/3, 5/3}, (a^3 z^3)/9])/ (9 3^(1/3) Gamma[4/3] Gamma[5/3])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["AiryAi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["z", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["1", "3"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", FractionBox["4", "3"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"]]], "9"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["2", "3"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["4", "3"], ",", FractionBox["5", "3"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"]]], "9"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["9", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]], ")"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <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> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ; </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 9 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[FractionBox[&quot;1&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;2&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;4&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[SuperscriptBox[&quot;a&quot;, &quot;3&quot;], &quot; &quot;, SuperscriptBox[&quot;z&quot;, &quot;3&quot;]]], &quot;9&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mroot> <mn> 3 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 5 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ; </mo> <mrow> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 9 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[FractionBox[&quot;2&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;4&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;5&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[SuperscriptBox[&quot;a&quot;, &quot;3&quot;], &quot; &quot;, SuperscriptBox[&quot;z&quot;, &quot;3&quot;]]], &quot;9&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <ci> AiryAi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 4 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 1 <sep /> 3 </cn> </list> <list> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 4 <sep /> 3 </cn> </list> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 4 <sep /> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 5 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 2 <sep /> 3 </cn> </list> <list> <cn type='rational'> 4 <sep /> 3 </cn> <cn type='rational'> 5 <sep /> 3 </cn> </list> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <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> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["AiryAi", "[", RowBox[List["a_", " ", "z_"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["1", "3"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", FractionBox["4", "3"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"]]], "9"]]], "]"]]]], RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]]]]], "-", FractionBox[RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["2", "3"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["4", "3"], ",", FractionBox["5", "3"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"]]], "9"]]], "]"]]]], RowBox[List["9", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]]]]]]]]]


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





2001-10-29