Borel transform
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| Borel transform | |
|---|---|
| Name | Borel transform |
| Type | Integral transform |
| Introduced | 1899 |
| Inventor | Émile Borel |
| Fields | Complex analysis; Asymptotic analysis; Quantum field theory |
Borel transform is an integral transform used to convert formal power series into analytic functions, often to study divergent series and enable resummation. It provides a bridge between formal asymptotic expansions and analytic continuation, playing a central role in summation methods, resurgence theory, and perturbative quantum field theory. The transform interacts with Laplace transforms, analytic continuation, and singularity analysis to recover functions from divergent expansions.
Definition
The Borel transform of a formal power series f(z)=∑_{n=0}^∞ a_n z^n is typically defined by mapping coefficients a_n to a new series or function B(f)(t)=∑_{n=0}^∞ a_n t^n / n!, which often has a larger radius of convergence. This construction can be viewed as a formal inverse to factorial growth: when a_n grow like n!, the Borel transform produces a function with finite radius of convergence. The transformed function can sometimes be identified with an analytic function on domains related to sectors studied by Émile Borel, Henri Poincaré, Gaston Darboux, Felix Klein, and Sofia Kovalevskaya.
Properties and Variants
Variants include the ordinary Borel transform, the accelerated Borel transform with factorial-generalized denominators, and the directional or sectoral Borel transform used in Écalle’s Jean Écalle resurgence theory. Key properties: linearity, coefficient-wise action on formal series, and compatibility with the Laplace transform under suitable growth conditions—a series with analytic Borel transform can be Laplace-integrated back. Relationships connect the transform to Mellin transforms studied by Rudolf Lipschitz, convolution algebras associated with Niels Henrik Abel, and operational calculus frameworks developed by Oliver Heaviside. Generalizations involve multisummability by Walter Balser and acceleration operators used by Heinrich Poincaré’s followers.
Borel summation and analytic continuation
Borel summation proceeds by taking the Borel transform, analytically continuing it along paths avoiding singularities, and applying a Laplace integral to reconstruct a summed value; this process was systematized by Émile Borel and extended by Jean Écalle, Ovidiu Costin, and David Sauzin. Analytic continuation of the Borel transform can reveal singularities, branch cuts, and Stokes phenomena linked to works of Sir Michael Berry, Sir Roger Penrose, and John von Neumann in mathematical physics contexts. Directional Borel summation across different rays in the complex plane produces Stokes discontinuities studied by Gian-Carlo Rota, Humberto Maturana, and contributors to resurgence describing alien calculus by Jean Écalle.
Applications in mathematics and physics
In pure mathematics, the transform is applied to ordinary and partial differential equations, singular perturbation problems, and the study of transseries as pursued by Dmitry Novikov, Jean-Pierre Ramis, and Michael Singer. In mathematical physics, it underpins perturbative expansions in quantum field theory associated with Richard Feynman, semiclassical expansions in quantum mechanics tied to Werner Heisenberg and Paul Dirac, and instanton calculus used by Gabriele Veneziano and Edward Witten. Borel techniques assist in summing divergent series arising in quantum chromodynamics connected to Murray Gell-Mann, renormalization group analyses linked to Kenneth Wilson, and string perturbation series appearing in Joseph Polchinski’s work. Further applications include analytic number theory problems related to Bernhard Riemann and functional equations studied by Andrey Kolmogorov.
Examples and computations
Classic examples include the factorial divergent series ∑ n! z^n whose Borel transform yields a geometric-type series convergent in a unit disc, enabling Laplace inversion to reconstruct a function with singularities determined by analytic continuation; such computations echo techniques from Augustin-Louis Cauchy and Karl Weierstrass. Airy- and Bessel-type asymptotic expansions appearing in papers by George Airy and Friedrich Bessel are commonly treated with Borel summation to connect asymptotic series to actual solutions of differential equations studied by Sofia Kovalevskaya and Henri Poincaré. Explicit calculations in quantum mechanics, such as divergent perturbation series for the anharmonic oscillator investigated by Lars Onsager and Tullio Regge, are resummed via Borel methods developed further by Jean Zinn-Justin and R. B. Dingle.
Historical background and development
The idea originated around the turn of the 20th century with Émile Borel’s foundational papers and was influenced by earlier analysis by Augustin-Louis Cauchy and Bernhard Riemann. Later systematic developments came from Henri Poincaré’s studies of asymptotic series and from 20th-century formalization by Émile Borel’s contemporaries and successors including Jean Écalle, Walter Balser, Ovidiu Costin, and David Sauzin. The method became integral to the mathematical formalism underlying perturbative quantum field theory as elaborated by Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga, and it continues to be refined in contemporary research bridging analysis and mathematical physics by groups around Max Planck Institute for Mathematics, Institute for Advanced Study, and major universities such as Princeton University and École Normale Supérieure.