Discrete frequency perturbation

In dealing with eigenmodes, we ususally encounter discrete frequency perturbations, i.e., $h (t)$ is periodic function of $t$ so that they contian only discrete frequency components. In this case, the inverse Fourier transformtion in Eq. (31) is replaced by the Fourier series, i.e.,

$$h (t) = \sum_{j = - \infty}^{\infty} c (\omega_j) e^{- i \omega_j t},$$ (38)

where the coefficients $c_j$ are given by

$$c (\omega_j) = \frac{1}{T} \int_0^T h (t) e^{i \omega_j t} d t,$$ (39)

with $\omega_j = j 2 \pi / T$ and $T$ being the period of $h (t)$ ($T$ is larger enough so that $1 / T$ is very small compared with frequency we are interested). The relation between $h (t)$ and $\hat{h} (\omega)$ given by Eqs. (33) and (34) also applies to the relation between $h (t)$ and $c (\omega)$, i.e.,

$$\int_{- \infty}^{\infty} \frac{\partial}{\partial t} h (t) e^{i \omega t} d t = - i \omega c (\omega)$$ (40)

and

$$\int_{- \infty}^{\infty} \frac{\partial^2}{\partial t^2} h (t) e^{i \omega t} d t = - \omega^2 c (\omega) .$$ (41)

Using the transformtion given by Eq. (39) on Eqs. (28), (29), and (30), respectively, we obtain the same equation as Eqs. (35)-(37).