Asymptotic birth trajectories in the discrete form of stable population theory

This paper seeks to supply a hitherto missing link in the reconciliation of the discrete and continuous formulations of stable population theory—see, for example, Keyfitz (“Introduction to the Mathematics of Population,” Chap. 8, Addison-Wesley, Reading, Mass., 1968). We show that the real exponential component of the birth trajectory has a non-zero (in fact positive) coefficient. Consequently, there is no possibility that oscillatory terms in the series expression for the birth trajectory can be dominant. This is shown by obtaining a formula for the desired coefficient.