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?



Limit for series of Bessel functions evaluated at zeros
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.


