Taylor series expansion

The operator on the RHS of Equation 4 can be expanded as a power series in $-i\omega$

$$\pmatrix{ I & 0 \cr 0 & I } - i \omega z \pmatrix{ ... ...omega^{2} z^{2}}{2} \pmatrix{ \rho s & 0 \cr 0 & \rho s }$$

+

 

$$i\frac{\omega^{3} z^{3}}{6} \pmatrix{ 0 & \rho s^{2} \cr ... ...}{24} \pmatrix{ \rho^{2} s^{2} & 0 \cr 0 & \rho^{2} s^{2} }$$ (8)

which may be considered as the Fourier transform of the first five terms of a Taylor series expansion of a time-domain convolutional kernel.