Fenton and Griffiths (2008) in Risk Assessment in Geotechnical Engineering give the following representation of a stochastic process:
$$Z(x_j) = \sum_{k=0}^{N} A_k \cos(\omega_k x_j) + B_k \sin(\omega_k x_j) $$
This can be rewritten as:
$$Z(x) = \sum_{k=0}^{N} (A_k - iB_k)e^{i\omega_kx_j}$$
Can anyone explain how to go from the first equation to the second? Do any assumptions/conclusions need to be made to do this?
My attempt:
\begin{align} A_k \cos(\omega_k x_j) + B_k \sin(\omega_k x_j) &= A_k\left(\frac{e^{i\omega_kt_j} + e^{-i\omega_kt_j}}{2}\right) + B_k\left(\frac{e^{i\omega_kt_j} - e^{-i\omega_kt_j}}{2}\right)\\ &= (A_k + B_k)\left(\frac{e^{i\omega_kt_j}}{2}\right) + (A_k - B_k)\left(\frac{e^{-i\omega_kt_j}}{2}\right) \end{align}