The accuracy of causal integration

The accuracy of the approximation of $\hat \omega$ to $\omega$can be seen by dividing the top and bottom of equation (13) by $\sqrt{Z}$and substituting $Z=e^{i\omega \Delta t}$:

$$\begin{eqnarray} -\,i \ { \hat \omega \, \Delta t \over 2} &=& {1\,-\,Z \over 1\... ...n \ { \omega \, \Delta t \over 2 } \ \hat \omega &\approx& \omega\end{eqnarray}$$ (19) (20) (21) (22)

This is a valid approximation at low frequencies.