Show that

by making a series expansion of the integrand. For C ≪ 1, that is, for (ω/k) ≪ (2k_BT_e/m_e)^1/2 , show that the dispersion relation for the longitudinal plasma wave reduces to

or

This result is the low-frequency limit of the result obtained from the macroscopic warm plasma model, using the isothermal sound speed of the electron gas

Transcribed Image Text:

∞ 20 To 20¹² (-1) n n=0 exp (W² - C²) dw 1x3x5x 2n 02n = (2n-1) (2n + 1)