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Transcendental Equation Help

  1. Dec 28, 2011 #1

    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.
  2. jcsd
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