相关论文: Variations on the Resurgence of the Gamma Function

A precise description of the singularities of the Borel transform of solutions of a level-one linear differential system is deduced from a proof of the summable-resurgence of the solutions by the perturbative method of J. \'Ecalle. Then we…

In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series,…

The theory of resurgence uniquely associates a factorially divergent formal power series with a collection of exponentially small non-perturbative corrections paired with a set of complex numbers known as Stokes constants. When the Borel…

The present text is an introduction to \'Ecalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for…

This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the…

For analytic nonlinear systems of ordinary differential equations, under some non-degeneracy and integrability conditions we prove that the formal exponential series solutions (trans-series) at an irregular singularity of rank one are Borel…

We compare Picard--Lefschetz theory and resurgence in three basic one-dimensional exponential integrals: the Airy model, the Bessel model, and the Gamma model. On the Picard--Lefschetz side, we describe the Lefschetz thimbles and compute…

We consider meromorphic transforms given by meromorphic kernels and study their asymptotic expansions under a certain rescaling. Under decay assumptions we establish the full asymptotic expansion in the rescaling parameter of these…

Given a holomorphic germ at the origin of C with a simple parabolic fixed point, the Fatou coordinates have a common asymptotic expansion whose formal Borel transform is resurgent. We show how to use Ecalle's alien operators to study the…

We consider partial theta series associated with periodic sequences of coefficients, of the form $\Theta(\tau) := \sum_{n>0} n^\nu f(n) e^{i\pi n^2\tau/M}$, with $\nu$ non-negative integer and an $M$-periodic function $f : \mathbb{Z}…

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…

In this thesis we explore the physics of renormalons in integrable models under the framework of resurgence. In the first part, we review some background on resurgence, integrability and renormalons, including a discussion of large N…

In many physical problems, it is important to capture exponentially-small effects that lie beyond-all-orders of a typical asymptotic expansion; when collected, the full expansion is known as the trans-series. Applied exponential asymptotics…

In a recent paper [arXiv:1406.1320] Paris has made several comments concerning the author's recent work on the exactification of Stirling's approximation for the logarithm of the gamma function, $\ln \Gamma(z)$. Despite acknowledging that…

In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions…

We provide an overview of the tools and techniques of resurgence theory used in the Borel-Ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of…

In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the…

We prove that the formal $\hbar$-power series solution of the deformed Painlev\'{e} I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find…

We consider a class of $n^{\text{th}}$-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of $u$. We demonstrate…

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series.…