  We give a direct derivation of the distribution of the maximum and the
location of the maximum of one-sided and two-sided Brownian motion with a
negative parabolic drift. The argument uses a relation between integrals of
special functions, in particular involving integrals with respect to functions
which can be called "incomplete Scorer functions". The relation is proved by
showing that both integrals, as a function of two parameters, satisfy the same
extended heat equation, and the maximum principle is used to show that these
solution must therefore have the stated relation. Once this relation is
established, a direct derivation of the distribution of the maximum and
location of the maximum of Brownian motion minus a parabola is possible,
leading to a considerable shortening of the original proofs.
