\[W_{k}\left(z\right)\sim\xi_{k}-\ln\xi_{k}+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{%
\xi_{k}^{n}}\sum_{m=1}^{n}\genfrac{[}{]}{0.0pt}{}{n}{n-m+1}\frac{\left(-\ln\xi%
_{k}\right)^{m}}{m!},\]