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$Assumptions = k >= 0 && k < k0 && k0 >= 0 && c >= 0 && n >= 0 ;
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f = Refine[Integrate[(1 - Exp[-c x])^\[Kappa] (k0 x)^(-q) Exp[
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I k0 x] x BesselJ[n, k x], {x,
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0, \[Infinity]}], {\[Kappa] == kk, q == qq, n == nn}]
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CForm[f]
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Series[f, {k, \[Infinity], 10}]
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Simplify[f]
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Quit[ ]
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