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In Eq. 2.27.15-2.27.17 ((,O,O) are spherical polar coordinates for k, with polar axis along Bo ; k/ = kcosO, k± = ksinO , and H(x) is the Heaviside step function. Eq. 2.27.14 can be regarded as an integral equation for f, and is a central result in the analysis.

By using the standard generating function identity for Bessel functions (e.g. Abramowitz and Stegun, M1965, p. 361, formula 9.1.41), Webb et al. (2001) obtained

xji - exp[-(( + v± + ik// rg cos9 + in)]} (2.27.18)

where rg = pc/ (qBo) is the particle gyro-radius and Jn (x) is the Bessel function of the first kind of order n and argument x. By noting that s = -/Q, and setting 5 = -im one finds that the denominator of the n-th term in Eq. 2.27.18

s + v± + ik//rg cos 9 + n = 0 when m - k//v/ = nQ - iv± (n integer), (2.27.19)

where / = cos 9. Thus the role for the term indexed by n in the series of Eq. 2.27.18 corresponds to the cyclotron resonance condition m- k//v/ = nQ broadened by scattering owed to v±. Averaging Eq. 2.27.14 over gyro-phase ( yields the integral equation fo = (o + Q ) + (1 - T///T± )a], (2.27.20)

~ ~ _ 2n relating fo and Fo where Q = JQd((2n). The function a in Eq. 2.27.20 can be

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