  In this note, we look at some of the less explored aspects of the gamma
function. We provide a new proof of Euler's reflection formula and discuss its
significance in the theory of special functions. We also discuss a result of
Landau concerning the determination of values of the gamma function using
functional identities. We show that his result is sharp and extend it to
complex arguments. In 1848, Oskar Schl\"omilch gave an interesting additive
analogue of the duplication formula. We prove a generalized version of this
formula using the theory of hypergeometric functions.
