We can then rewrite the ESKC equation as

$$ \frac{\partial W(\mathbf{x},s|\mathbf{x}_0)}{\partial s}\tau=-W(\mathbf{x},s|\mathbf{x}_0)+ \sum_{n=0}^{\infty}\frac{(-\xi)^n}{n!}\frac{\partial^n}{\partial x^n} \left[W(\mathbf{x},s|\mathbf{x}_0)\int_{-\infty}^{\infty} \xi^nW(\mathbf{x}+\xi,\tau|\mathbf{x})d\xi\right]. $$

We have neglected higher powers of \( \tau \) and have used that for \( n=0 \) we get simply \( W(\mathbf{x},s|\mathbf{x}_0) \) due to normalization.