A power series expansion for the truncated lognormal characteristic function

An infinite series expansion for the characteristic function of the lognormal distribution does not exist, and no infinite series representation of the characteristic function of any modified form of the lognormal distribution is found in the literature. A power series expansion is derived for the characteristic function of the truncated lognormal distribution. The series is proved to converge absolutely for any level of truncation. Equivalently, the series converges absolutely for any nonzero value of probability in the missing tail, and the truncated lognormal can be made arbitrarily close, but not equal to, the lognormal while retaining convergence. The behaviours of the moments of the truncated lognormal and lognormal distributions are examined in detail.