# Limit for series of Bessel functions evaluated at zeros

The following series arises in an electrostatics problem for a conducting cylinder: $$V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}$$ where $$J_i$$ is the Bessel function of $$i^{th}$$ order, and $$k_n$$ is the location of the $$n^{th}$$ zero of $$J_0$$. $$V$$ can be proven to converge for $$z>0$$, and from numerical tests, converges conditionally also at $$z=0$$ except for poles at $$\rho=0,2,4,6...$$

Is there any kind of analytic or asymptotic expression as a function of $$\rho$$ in the limit as $$z\rightarrow0$$, for $$\rho>2$$ in particular?

• In relation to your question, it seems that there would potentially be something helpful in Watson's Bessel functions book. I can't say for sure though. – JCM Jan 18 at 23:03

## 1 Answer

Employing the asymptotics of large zeroes of Bessel functions and the large-argument asymptotics of the Bessel functions, it can be shown that the $$n$$th term of the series behaves like $$\frac{1}{\sqrt{2n\rho}}\cos\left(\rho\left(n-\tfrac{1}{4} \right)\pi-\tfrac{\pi}{4}\right)e^{ -\left(n-\tfrac{1}{4}\right)\pi z} + \mathcal{O}_{z,\rho}\left(\frac{1}{n^{3/2}}\right).$$ Thus, in terms of the polylogarithm, $$V = \frac{e^{\frac{\pi}{4}z}}{\sqrt {2\rho }}\Re \left[ e^{ - \frac{\pi}{4}(\rho + 1)i} \operatorname{Li}_{1/2} \left( e^{\pi \rho i - \pi z} \right) \right] + \mathcal{O}_{z,\rho } (1).$$ Yo may proceed from here.