  We provide estimates for the convolution product of an arbitrary number of
"resurgent functions", that is holomorphic germs at the origin of $C$ that
admit analytic continuation outside a closed discrete subset of $C$ which is
stable under addition. Such estimates are then used to perform nonlinear
operations like substitution in a convergent series, composition or functional
inversion with resurgent functions, and to justify the rules of "alien
calculus"; they also yield implicitly defined resurgent functions. The same
nonlinear operations can be performed in the framework of Borel-Laplace
summability.
