Report for last 10 days.

I've completed my article. But at the last moment I've found out that my infinite series representation for Bessel functions can be analytically continued to the entire complex plane! Now I have to prove it rigorously and rewrite some pieces of article.

Among other I need to prove that following representation for Bessel function $J_1(z)$ where $z\in\mathbb{C}$ is true: 
\begin{equation}
\label{eq:J1}
J_1(z)=\frac{2}{z}\sum_{k=0}^{\infty}\varepsilon_k\big[(-1)^k-\cos\sqrt{z^2+(k\pi)^2}\big]
\end{equation}
where\begin{equation}
\label{eq:eps}
\varepsilon_k=
\begin{cases}
\frac12, & \mbox{if } k=0 \\
1, & \mbox{if } k>0.
\end{cases}
\end{equation}