As part of a derivation in Bernstein '58 [1] a linear first-order (eqn. (9) in the image) appears:
But the general solution I would usually take (as appears in Gradshteyn and Ryzhik and checked in Mathematica) here would be
$$F=c_1G+\frac1G\int^\phi_1\frac{G(\phi^\prime)\Phi(\phi^\prime)}\Omega$$ where $\Phi$ is the RHS of eqn. (9). I suppose it depends on the requirement of periodicity and that $s$ has positive complex part (it comes from a Laplace transform) but I can't see how Bernstein's solution is obtained or equivalent, i.e. where the factor of $1/G$ goes. I'm sure I'm missing something silly, but I've wasted a bit of time on this already and can't get it to work out. The same derivation appears in slightly neater form in Montgomery and Tidman[2] but without further insight. Any help justifying this result would be greatly appreciated.
[1] Bernstein, I. (1958). Waves in a plasma in a magnetic field. Physical Review. Retrieved from http://journals.aps.org/pr/abstract/10.1103/PhysRev.109.10
[2] Montgomery D.C., Tidman, D.A. (1964). Plasma Kinetic Theory, McGraw-Hill advanced physics monograph series
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