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http://functions.wolfram.com/03.01.21.0081.01
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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
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</mrow> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> > </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> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> μ </ci> <apply> 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<apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> b </ci> <ci> a </ci> </apply> <apply> <lt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <ci> ν </ci> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> </apply> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> a </ci> <ci> b </ci> </apply> <apply> <lt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> b </ci> <ci> a </ci> </apply> <apply> <lt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </piece> <otherwise> <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> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> μ </ci> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> </apply> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <and /> <apply> <gt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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