Mould expansions for the saddle-node and resurgence monomials

This article is an introduction to some aspects of Ecalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler's vector field x(2) partial derivative/partial derivative x + (x + y) partial derivative/partial derivative y, and for which the formal normalisation is shown to be resurgent in 1/x. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus.