In the present work, we carried out a study on the Lambert function W which is implicitly de-
fined as the inverse of the function f(x) = x expx. We start by constructing the graph of f and
then reflect it around the line y = x in order to obtain the graph of W. We make a detailed pre-
sentation of the particularities of its formation law, the graphic outline highlighting its domain
and image, since it is a multivalued function, that is, defined in branches. We present some of
its immediate properties and fundamental identities, beyond the principle of simplification. We
display some notable and special values of the function, such as the Omega constant. We study
the applications of the W function in obtaining solutions to transcendental equations. Finally,
we also study the infinitesimal and integral calculus of Lambert W function.
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