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In mathematics, Borel summation is a summation method for divergent series, introduced by Émile Borel (1899). There are several variations of this method that are also called Borel summation.

Definition

Let

$$y(z) = \sum_{k = 0}^\infty y_kz^k$$

be a formal power series in z.

Define the Borel transform $$\mathcal{B}y$$ of y by

$$\mathcal{B}y(t) := \sum_{k=0}^\infty \frac{y_k}{k!}t^k.$$

Suppose that the Borel transform converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Borel sum of y is given by

\int_0^\infty e^{-t} \mathcal{B}y(tz) \, dt. \)

A slightly weaker form of Borel's summation method gives the Borel sum of y as

\lim_{t\rightarrow\infty} e^{-t}\sum_n\frac{t^n}{n!}\cdot\sum_{k\le n}y_kz^k \)

If the sum exists in this sense then it also exists in the previous sense and is the same, but there are some series that can be summed with the previous method but not with this method (Hardy 1992, 8.5).
Examples

The series

$$y(z) = \sum_{k = 0}^\infty z^k$$

converges to 1/(1 − z) for |z| < 1. The Borel transform is

$$\mathcal{B}y(t) := \sum_{k=0}^\infty \frac{1}{k!}t^k = e^t.$$

So the Borel sum is

$$\int_0^\infty e^{-t}\mathcal{B}y(tz) \, dt = \int_0^\infty e^{-t} e^{tz} \, dt =\frac{1}{1-z}.$$

which converges to 1/(1 − z) in the larger region Re(z) < 1, giving an analytic continuation of the original series.

The series

$$y(z) = \sum_{k = 0}^\infty (-1)^kk!z^k$$

does not converge for any nonzero z.

The Borel transform is

$$\mathcal{B}y(t) := \sum_{k=0}^\infty (-1)^kt^k = \frac{1}{1+t}$$

for |t| < 1, and this can be analytically continued to all t ≥ 0. So the Borel sum is

\int_0^\infty e^{-t}\mathcal{B}y(tz) \, dt = \int_0^\infty \frac{e^{-t}} {1+tz} \, dt = \frac 1 z \cdot e^\frac 1 z \cdot \Gamma\left(0,\frac 1 z \right) \)

(where Γ is the incomplete Gamma function).

This integral converges for all z ≥ 0, so the original divergent series is Borel summable for all such z. This function has an asymptotic expansion as z tends to 0 that is given by the original divergent series. This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptotic expansions.
The Borel polygon

The region where the power series of an analytic function is Borel summable was described as follows by Borel and Phragmen (Sansone & Gerretsen 1960, 8.3).

If y is a power series that converges in some neighborhood of the origin then it has a Borel sum at some point z if it can be analytically continued to a disc with diameter 0z. Conversely if the function can be analytically continued to the disc with diameter 0z then it is Borel summable at z.

The set of points z such that the function can be analytically continued to the interior of the disk with diameter 0z is a polygon when the function has only a finite number of singularities, called the Borel polygon. Its edges pass through the singular points and are orthogonal to lines joining the singular points to 0.
Applications

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).
Notes

Divergent series
Euler summation
Cesàro summation
Lambert summation
Nachbin resummation
Abelian and tauberian theorems
Van Wijngaarden transformation

References

Borel, E. (1899), "Mémoire sur les séries divergentes", Ann. Sci. École Norm. Sup. (3) 16: 9–131
Glimm, James; Jaffe, Arthur (1987), Quantum physics (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96476-8, MR887102
Hardy, Godfrey Harold (1992) [1949], Divergent Series, New York: Chelsea, ISBN 978-0-8218-2649-2, MR0030620
Reed, Michael; Simon, Barry (1978), Methods of modern mathematical physics. IV. Analysis of operators, New York: Academic Press [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-585004-9, MR0493421
Sansone, Giovanni; Gerretsen, Johan (1960), Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen, MR0113988
Weinberg, Steven (2005), The quantum theory of fields. Vol. II, Cambridge University Press, ISBN 978-0-521-55002-4, MR2148467
Zakharov, A. A. (2001), "Borel summation method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104