Conditional convergence of J1(kr)k

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SUMMARY

The discussion centers on the integration of the oscillatory function \(\int_0^{\infty} J_1(k r)k dk\), where \(J_1(x)\) is the cylindrical Bessel function of the first kind. René Repetit proposes using the technique of multiplying the integrand by \(e^{-kd}\) to facilitate convergence, but Jason expresses skepticism about the convergence of the integral itself. Jason highlights the need to prove convergence before attempting to evaluate the integral, suggesting that the series formed by the integral does not converge due to the limit of its terms not approaching zero.

PREREQUISITES
  • Understanding of cylindrical Bessel functions, specifically \(J_1(x)\).
  • Knowledge of oscillatory integrals and their convergence properties.
  • Familiarity with techniques for evaluating improper integrals.
  • Basic concepts of series convergence and divergence.
NEXT STEPS
  • Research the properties and applications of cylindrical Bessel functions, particularly \(J_1(x)\).
  • Study techniques for evaluating oscillatory integrals, including the use of damping factors like \(e^{-kd}\).
  • Learn about convergence tests for series, focusing on alternating series.
  • Explore advanced integration techniques, such as contour integration in complex analysis.
USEFUL FOR

Mathematicians, physicists, and engineers dealing with oscillatory integrals, particularly those interested in the properties of Bessel functions and convergence analysis.

Repetit
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Hey!

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.

René
 
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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 [tex]a_n[/tex] denote the [tex]n^{th}[/tex] term in the series. Since [tex]lim_{n\rightarrow \infty} a_n \neq 0[/tex] 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
 

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