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
 











AiryBiPrime






Mathematica Notation

Traditional Notation









Bessel-Type Functions > AiryBiPrime[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function and a power function > Involving exp and power > Linear arguments





http://functions.wolfram.com/03.08.21.0014.01









  


  










Input Form





Integrate[(Sqrt[z] AiryBiPrime[a z])/E^((2/3) (a z)^(3/2)), z] == (1/(21 a^2 Sqrt[z] Gamma[1/3])) ((2 (3 a^2 z^2 AiryBiPrime[a z] Gamma[1/3] + 3^(1/6) Sqrt[a z] (6 E^((2/3) (a z)^(3/2)) + (3^(1/3) a^3 z^3 BesselI[-(5/3), (2/3) a^(3/2) z^(3/2)] Gamma[1/3])/(a^(3/2) z^(3/2))^(1/3) + 3^(1/3) a^2 z^2 (a^(3/2) z^(3/2))^(1/3) BesselI[5/3, (2/3) a^(3/2) z^(3/2)] Gamma[1/3])))/E^((2/3) (a z)^(3/2)))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox["z"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["AiryBiPrime", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["21", " ", SuperscriptBox["a", "2"], " ", SqrtBox["z"], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]], RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SqrtBox[RowBox[List["a", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", FractionBox["5", "3"]]], ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["BesselI", "[", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "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> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> Bi </mi> <mo> &#8242; </mo> </msup> <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> <mfrac> <mn> 1 </mn> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> Bi </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <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> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mroot> <mn> 3 </mn> <mn> 6 </mn> </mroot> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mroot> <mn> 3 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mfrac> <mn> 5 </mn> <mn> 3 </mn> </mfrac> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <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> <mroot> <mrow> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mroot> <mn> 3 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mroot> <mrow> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mn> 3 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mfrac> <mn> 5 </mn> <mn> 3 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> AiryBiPrime </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> AiryBiPrime </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> BesselI </ci> <cn type='rational'> 5 <sep /> 3 </cn> <apply> <times /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 3 </cn> </apply> <apply> <times /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> </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[SqrtBox["z_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", "z_"]], ")"]], RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SqrtBox[RowBox[List["a", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", FractionBox["5", "3"]]], ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], ")"]], RowBox[List["1", "/", "3"]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["BesselI", "[", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["21", " ", SuperscriptBox["a", "2"], " ", SqrtBox["z"], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]]]]]]










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





2001-10-29