1. A. R. Forsyth,A Treatise on Differential Equations(MacMillan and Company, Ltd., London, 522, 1914).
2. Waves in a Plasma in a Magnetic Field
3. Excitation of Longitudinal Plasma Oscillations Near Electron Cyclotron Harmonics
4. The derivation is not difficult but tedious. Equation (7) is written as ∇Tϕ2 = −en0ɛ00∞ f1u du 02π dψ −∞∞ dvz. Intergration over vz is trivial when kz = 0. Integration over ψ gives a realation betweenk,l,m, andnwhich insures that the intergration does not vanish. The result is ∇Tϕ2 = ωp2ωc l=−∞∞ m=−∞∞ n=−∞∞ 0∞ Il(bx)Im(bx)⋅Jn(by) (−1)la−n−m −Jʹn−m−1(by) ∂ϕ∂x g+n−m−lbyJn−m−l(by)(−j ∂ϕ∂y g)⋅u2meT2 exp −u22eT∕m du. Since the lowest order of Jn(x) is xn, the power series expansion ofuof the above sums of the products of Bessel functions has the form, a1u+a3u3+⋯. The expansion up to the term u3 can be carried out by taking the combination ofl,mandnsuch that ∣l∣+∣m∣+∣n∣+∣n−m−l±1∣≤3. The choice of sign in the above expression is governed by convenience.