tmp -- -to be deleted

-- -- -- -- As is given in the wikipedia, the ponderomotive force is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field. The ponderomotive force $\mathbf{F}_p$ is expressed by

$$\mathbf{F}_p = - \frac{e^2}{4 m \omega^2} \nabla E^2,$$ (94)

where $e$ is the electrical charge of the particle, $m$ is the mass of the particle, $E$ is the amplitude of the inhomogeneous oscillating electric field (at low enough amplitudes the magnetic field exerts very little force), $\omega$ is the angular frequency of oscillation of the field. to be continued --

$$\varepsilon (k, \omega) = 1 + \frac{e^2}{\varepsilon_0 m_e k} \int_C \frac{\partial f_0 / \partial v}{\omega - k v} d v = 0$$

$$1 - 2 \frac{e^2}{\varepsilon_0 m_e k} \frac{n_0}{v_t \sqrt{\pi}} \frac{1}{k v_t} \int_C \frac{\exp (- t^2)}{\zeta - t} t d t = 0$$

$$Z (\zeta) = 2 i e^{- \zeta^2} \int_{- \infty}^{i \zeta} e^{- t^2} d t.$$ (95)

$$Z (\zeta) = i \sqrt{\pi} w (\zeta),$$ (96)

where $w (\zeta)$ is Faddeeva's function, which is defined by

$$w (\zeta) = \exp (- \zeta^2) \ensuremath{\operatorname{erfc}} (- i \zeta) .$$ (97)

$$Z (\zeta) = i \sqrt{\pi} \exp (- \zeta^2) \ensuremath{\operatorname{erfc}} (- i \zeta)$$

$$= i \sqrt{\pi} \exp (- \zeta^2) \frac{2}{\sqrt{\pi}} \int_{- i \zeta}^{\infty} e^{- t^2} d t$$

$$= 2 i \exp (- \zeta^2) \int_{- \infty}^{i \zeta} e^{- t^2} d t$$ (98)