On the Asymptotic Form of the Collision Density

In a single-species system with similarly varying cross sections, it is commonly assumed that the collision density F(u) has the asymptotic form ke^mu, where m satisfies the equation (1 − α) (1 + m) − c(1 − α^1+m) = 0. This is equivalent to assuming that the pole with greatest real part of the Laplace transform of F(u) occurs at the real root m(≠−1) of the last equation. No proof of this assumption appears to have been given hitherto in the literature, so it is now shown, by the use of certain results in the theory of transcendental equations, that if z is any complex root of the equation, then irrespective of the values of α and c, Re z < min (−1, m). Finally, the constant k in the assumed form of F(u) is determined exactly, in terms of m, by taking the residue at m of the Laplace transform of F(u).