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.
-resurgent function is an element of
δ ⊕
, i.e. an element of the form
δ ⊕
[2] A power series
whose formal Borel transformation is a
-resurgent function is called
-resurgent series.
: The formal power series
if the associated formal power series
ψ ( t ) = ϕ ( 1
has a positive radius of convergence.
denotes the space of formal power series convergent at
[2] Formal Borel transform: The formal Borel transform (named after Émile Borel) is the operator
defined by Convolution in
, then the convolution is given by By adjunction we can add a unit to the convolution in
and introduce the vector space
element with
we can write the space as
and define and set
be a non-empty discrete subset of
be the radius of convergence of
analytic continuation along some path in
starting at a point in
denotes the space of