Boreltransform

The Borel transform is a mathematical operator used primarily in the theory of divergent series and in complex analysis. Given a formal power series \(f(z)=\sum_{n=0}^{\infty}a_nz^n\), the Borel transform is defined by \(\mathcal{B}\{f\}(t)=\sum_{n=0}^{\infty}\frac{a_n}{n!}t^n\). The resulting series typically has a larger radius of convergence than the original. If the Borel transform converges on a suitable domain, it can be analytically continued and then integrated along a contour in the complex plane to recover a sum for the original divergent series. This procedure is known as Borel summation.

The transform was introduced by the French mathematician Émile Borel in the early 20th century as a technique for assigning values to divergent series that arise, for example, in perturbation expansions in quantum mechanics and in the asymptotic analysis of solutions to differential equations. Unlike the Laplace transform, which integrates the entire function, the Borel transform uses factorial weighting to tame growth of coefficients.

Key properties include linearity, compatibility with differentiation and integration (up to constants), and the fact that the Borel transform of a convergent power series reproduces the original series when the inverse Laplace transform is applied. In practice, many divergent perturbative series in physics are rendered Borel-summable after applying analytic continuation techniques to the Borel transform. Numerical implementations of Borel summation often employ Padé approximants to improve convergence before performing the Laplace integral. The method remains a central tool in resurgence theory, providing a bridge between formal series, analytic functions, and physical observables.