Conditional convergence of J1(kr)k

[Repetit]

Hey!

I need to perform the following integration:

$$\int\limits_0^{\infty} J_1(k r)k dk$$

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

I hope someone has a suggestion.

René

 

[jasonRF]

Repetit

Are you sure this integral converges? I am skeptical. If you think about the integral as a sequence of integrals between each zero crossing, you need to sum an alternating series. Let $$a_n$$ denote the $$n^{th}$$ term in the series. Since $$lim_{n\rightarrow \infty} a_n \neq 0$$ the series doesn't converge. This isn't really a proof - I leave that to you. But I think you need to prove that this integral converges before you spend much time trying to evaluate it!

Jason