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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/03.01.21.0081.01









  


  










Input Form





Integrate[t^(\[Alpha] - 1) BesselJ[\[Mu], a t] BesselJ[\[Nu], b t], {t, 0, Infinity}] == Piecewise[ {{((2^(\[Alpha] - 1) b^(-\[Mu] - \[Alpha]) a^\[Mu] Gamma[(\[Alpha] + \[Mu] + \[Nu])/2])/ (Gamma[(\[Nu] - \[Mu] - \[Alpha])/2 + 1] Gamma[\[Mu] + 1])) Hypergeometric2F1[(\[Alpha] + \[Mu] + \[Nu])/2, (\[Mu] - \[Nu] + \[Alpha])/2, \[Mu] + 1, a^2/b^2], b > a && Re[\[Alpha]] < 2}, {((2^(\[Alpha] - 1) a^(-\[Nu] - \[Alpha]) b^\[Nu] Gamma[(\[Alpha] + \[Mu] + \[Nu])/2])/ (Gamma[(\[Mu] - \[Nu] - \[Alpha])/2 + 1] Gamma[\[Nu] + 1])) Hypergeometric2F1[(\[Alpha] + \[Mu] + \[Nu])/2, (\[Nu] - \[Mu] + \[Alpha])/2, \[Nu] + 1, b^2/a^2], a > b && Re[\[Alpha]] < 2}, {(2^(\[Alpha] - 1) Gamma[(\[Alpha] + \[Mu] + \[Nu])/2] Gamma[1 - \[Alpha]])/(a^\[Alpha] (Gamma[(\[Mu] - \[Nu] - \[Alpha])/2 + 1] Gamma[1 + (1/2) (-\[Alpha] - \[Mu] + \[Nu])] Gamma[1 + (1/2) (-\[Alpha] + \[Mu] + \[Nu])])), b == a && Re[\[Alpha]] < 1}}, Integrate[t^(\[Alpha] - 1) BesselJ[\[Mu], a t] BesselJ[\[Nu], b t], {t, 0, Infinity}]] /; a > 0 && b > 0 && Re[\[Alpha] + \[Mu] + \[Nu]] > 0










Standard Form





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MathML Form







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</mo> <mrow> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> &#956; </ci> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <piecewise> <piece> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; 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Rule Form





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





2007-05-02





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