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I need to perform the following integration:

[tex]\int\limits_0^{\infty} J_1(k r)k dk[/tex]

where [tex]J_1(x)[/tex] is the cylindrical Bessel function of the first kind. This is an oscillatory function with amplitude decreasing as [tex]1/\sqrt{x}[/tex]. Due to the multiplication of [tex]k[/tex] the integrand however, becomes an oscillatory function which increases as a function of [tex]x[/tex]. How can I perform this integration? I was thinking about multiplying the integrand by [tex]e^{-k d}[/tex], performing the integration and then taking the limit as [tex]d[/tex] approaches zero, but I can't figure out how to evaluate the integral.

I hope someone has a suggestion.

RenĂ©

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# Conditional convergence of J1(kr)k

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