\[\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t=\frac{1}{2}\sum_{k=1}%
^{\infty}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}%
J_{k}\left(x\right),\]