Fourier expansion over $\theta$ and $\zeta$

A perturbation $G (\psi, \theta, \varphi)$ must be a periodic function of the poloidal angle $\theta$ and toroidal angle $\zeta$, and thus can be expanded as the following two-dimensional Fourier series,

$$G (\psi, \theta, \zeta) = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} G_{n m} (\psi) e^{i (m \theta - n \zeta)},$$ (170)

where the expansion coefficient $G_{n m}$ is given by

$$G_{n m} (\psi) = \frac{1}{(2 \pi)^2} \int_0^{2 \pi} \int_0^{2 \pi} G (\psi, \theta, \zeta) e^{i (n \zeta - m \theta)} d \theta d \zeta .$$ (171)

Our next task is to derive the equations that the coefficients $G_{n m}$ must satisfy.