\[w\left(z\right)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\,%
\mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\,\mathrm{d%
}t}{t^{2}-z^{2}},\]