  We study the analytical properties of the Laplace transform of the lognormal
distribution. Two integral expressions for the analytic continuation of the
Laplace transform of the lognormal distribution are provided, one of which
takes the form of a Mellin-Barnes integral. As a corollary, we obtain an
integral expression for the characteristic function; we show that the integral
expression derived by Leipnik in [11] is incorrect. We present two
approximations for the Laplace transform of the lognormal distribution, both
valid in $\mathbb{C} \setminus(-\infty,0]$. In the last section, we discuss how
one may use our results to compute the density of a sum of independent
lognormal random variables.
