\[\int_{0}^{z}\mathbf{K}_{0}\left(t\right)\,\mathrm{d}t-\frac{2}{\pi}(\ln\left(2%
z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(2k)!(2k-1%
)!}{(k!)^{2}(2z)^{2k}},\]