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I am reading a paper where part of the solution to an equation of diffusion in a multicompartment system includes the "sum of all nonnegative roots kj of the following transcendental equation,

2*u*(cos(k)-cos(q))-k*sin(k) = 0.

Then the authors of the paper say: "Note that the roots periodically depend on the parameter q, and, for their analysis, it is sufficient ot consider 0<q<pi. In the case of small u (i.e. uMM1), all roots are close to j*pi:

k0 approximately equals 2*(u^0.5)*sin(q/2), j=0

kj approximately equals j*pi + ((2*u)/(j*pi))*(1-(-1)^j cos(q))

end quote.

Can somebody help me with this derivation? The whole statement about roots periodically depending on q is confusing because q is fixed constant of the system. Further it seems to me that if we consider the case were u<<1, then the equation simplifies to -ksin(k) = 0 whith the roots not having q or u in them at all.

What am I missing????

Thanks for any help.