Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series.
He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis.
Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.
Suppose that the Borel transform converges for all positive real numbers to a function growing sufficiently slowly that the following integral is well defined (as an improper integral), the Borel sum of A is given by representing Laplace transform of
i.e. Regularity of (B) is easily seen by a change in order of integration, which is valid due to absolute convergence: if A(z) is convergent at z, then where the rightmost expression is exactly the Borel sum at z. Regularity of (B) and (wB) imply that these methods provide analytic extensions to A(z).
Watson's theorem gives conditions for a function to be the Borel sum of its asymptotic series.
Suppose that f is a function satisfying the following conditions: is bounded by for all z in the region (for some positive constant C).
Then Watson's theorem says that in this region f is given by the Borel sum of its asymptotic series.
More precisely, the series for the Borel transform converges in a neighborhood of the origin, and can be analytically continued to the positive real axis, and the integral defining the Borel sum converges to f(z) for z in the region above.
Carleman's theorem shows that a function is uniquely determined by an asymptotic series in a sector provided the errors in the finite order approximations do not grow too fast.
More precisely it states that if f is analytic in the interior of the sector |z| < C, Re(z) > 0 and |f(z)| < |bnz|n in this region for all n, then f is zero provided that the series 1/b0 + 1/b1 + ... diverges.
Carleman's theorem gives a summation method for any asymptotic series whose terms do not grow too fast, as the sum can be defined to be the unique function with this asymptotic series in a suitable sector if it exists.
Borel summation is slightly weaker than special case of this when bn =cn for some constant c. More generally one can define summation methods slightly stronger than Borel's by taking the numbers bn to be slightly larger, for example bn = cnlog n or bn =cnlog n log log n. In practice this generalization is of little use, as there are almost no natural examples of series summable by this method that cannot also be summed by Borel's method.
Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since for Re(z) < 1, Consider the series then A(z) does not converge for any nonzero z ∈ C. The Borel transform is for |t| < 1, which can be analytically continued to all t ≥ 0.
This integral converges for all z ≥ 0, so the original divergent series is Borel summable for all such z.
This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptotic expansions.
Again, since for all z, the equivalence theorem ensures that weak Borel summation has the same domain of convergence, z ≥ 0.
Suppose that A(z) has strictly positive radius of convergence, so that it is analytic in a non-trivial region containing the origin, and let SA denote the set of singularities of A.
For P ∈ SA, let LP denote the line passing through P which is perpendicular to the chord OP.
It can however be shown[2] that A does not converge for any point z ∈ C such that z2n = 1 for some n. Since the set of such z is dense in the unit circle, there can be no analytic extension of A outside of B(0,1).
Subsequently the largest star domain to which A can be analytically extended is S = B(0,1) from which (via the second definition) one obtains
Borel summation finds application in perturbation expansions in quantum field theory.
In particular in 2-dimensional Euclidean field theory the Schwinger functions can often be recovered from their perturbation series using Borel summation (Glimm & Jaffe 1987, p. 461).
Some of the singularities of the Borel transform are related to instantons and renormalons in quantum field theory (Weinberg 2005, 20.7).