\[\mathbf{H}_{1}\left(z\right)=\frac{2}{\pi}(1-J_{0}\left(z\right))+\frac{4}{\pi%
}\sum_{k=1}^{\infty}\frac{J_{2k}\left(z\right)}{4k^{2}-1}=4\sum_{k=0}^{\infty}%
J_{2k+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\tfrac{1}{%
2}z\right).\]