The term resurgent function (from Latin: resurgere, to get up again) comes from French mathematician Jean Écalle's theory of resurgent functions and alien calculus. The theory evolved from the summability of divergent series (see Borel summation) and treats analytic functions with isolated singularities. He introduced the term in the late 1970s.[1]
Resurgent functions have applications in asymptotic analysis, in the theory of differential equations, in perturbation theory and in quantum field theory.
For analytic functions with isolated singularities, the Alien calculus can be derived, a special algebra for their derivatives.