Borel summation

In mathematics, Borel summation is a summation method for divergent series, introduced by Template:Harvs. It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.

Definition

There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer.

Throughout let Template:Math denote a formal power series

$A(z)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k,$

and define the Borel transform of Template:Math to be its corresponding exponential series

$ℬA(t)\equiv \sum _{k=0}^{\mathrm{\infty }}\frac{a_k}{k!}{t}^k.$

Borel's exponential summation method

Let Template:Math denote the partial sum

$A_n(z)=\sum _{k=0}^n{a}_k{z}^k.$

A weak form of Borel's summation method defines the Borel sum of Template:Math to be

$\underset{t\to \mathrm{\infty }}{\lim }{e}^{-t}\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{n!}{A}_n(z).$

If this converges at Template:Math to some function Template:Math, we say that the weak Borel sum of Template:Math converges at Template:Math, and write $\sum a_k{z}^k=a(z)(𝒘𝑩)$.

Borel's integral summation method

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 Template:Math is given by

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}ℬA(tz)dt,$

representing Laplace transform of $ℬA(z)$.

If the integral converges at Template:Math to some Template:Math, we say that the Borel sum of Template:Math converges at Template:Math, and write $\sum a_k{z}^k=a(z)(𝑩)$.

Borel's integral summation method with analytic continuation

This is similar to Borel's integral summation method, except that the Borel transform need not converge for all Template:Math, but converges to an analytic function of Template:Math near 0 that can be analytically continued along the positive real axis.

Basic properties

Regularity

The methods Template:Math and Template:Math are both regular summation methods, meaning that whenever Template:Math converges (in the standard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.

$\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k=A(z)<\mathrm{\infty }\Rightarrow \sum a_k{z}^k=A(z)(𝑩,𝒘𝑩).$

Regularity of Template:Math is easily seen by a change in order of integration, which is valid due to absolute convergence: if Template:Math is convergent at Template:Math, then

$A(z)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k=\sum _{k=0}^{\mathrm{\infty }}{a}_k\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{t}^{k}dt\right)\frac{z^k}{k!}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}\sum _{k=0}^{\mathrm{\infty }}{a}_k\frac{(tz{)}^k}{k!}dt,$

where the rightmost expression is exactly the Borel sum at Template:Math.

Regularity of Template:Math and Template:Math imply that these methods provide analytic extensions to Template:Math.

Nonequivalence of Borel and weak Borel summation

Any series Template:Math that is weak Borel summable at Template:Math is also Borel summable at Template:Math. However, one can construct examples of series which are divergent under weak Borel summation, but which are Borel summable. The following theorem characterises the equivalence of the two methods.

Theorem (Template:Harv).

Let Template:Math be a formal power series, and fix Template:Math, then:

1. If $\sum a_k{z}^k=a(z)(𝒘𝑩)$, then $\sum a_k{z}^k=a(z)(𝑩)$.
2. If $\sum a_k{z}^k=a(z)(𝑩)$, and $\underset{t\to \mathrm{\infty }}{\lim }{e}^{-t}ℬA(zt)=0,$ then $\sum a_k{z}^k=a(z)(𝒘𝑩)$.

Relationship to other summation methods

• Template:Math is the special case of Mittag-Leffler summation with Template:Math.
• Template:Math can be seen as the limiting case of generalized Euler summation method Template:Math in the sense that as Template:Math the domain of convergence of the Template:Math method converges up to the domain of convergence for Template:Math.

Uniqueness theorems

There are always many different functions with any given asymptotic expansion. However, there is sometimes a best possible function, in the sense that the errors in the finite-dimensional approximations are as small as possible in some region. Watson's theorem and Carleman's theorem show that Borel summation produces such a best possible sum of the series.

Watson's theorem

Watson's theorem gives conditions for a function to be the Borel sum of its asymptotic series. Suppose that Template:Math is a function satisfying the following conditions:

• Template:Math is holomorphic in some region Template:Math, Template:Math for some positive Template:Math and Template:Math.
• In this region Template:Math has an asymptotic series Template:Math with the property that the error

$|f(z)-a_0-a_{1}z-\cdots -a_{n-1}{z}^{n-1}|$

is bounded by

$C^{n+1}n!|z{|}^n$

for all Template:Math in the region (for some positive constant Template:Math).

Then Watson's theorem says that in this region Template:Math 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 Template:Math for Template:Math in the region above.

Carleman's theorem

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 Template:Math is analytic in the interior of the sector Template:Math, Template:Math and Template:Math in this region for all Template:Math, then Template:Math is zero provided that the series Template:Math 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 Template:Math for some constant Template:Math. More generally one can define summation methods slightly stronger than Borel's by taking the numbers Template:Math to be slightly larger, for example Template:Math or Template:Math. 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.

Example

The function Template:Math has the asymptotic series Template:Math with an error bound of the form above in the region Template:Math for any Template:Math, but is not given by the Borel sum of its asymptotic series. This shows that the number Template:Math in Watson's theorem cannot be replaced by any smaller number (unless the bound on the error is made smaller).

Examples

The geometric series

Consider the geometric series

$A(z)=\sum _{k=0}^{\mathrm{\infty }}{z}^k,$

which converges (in the standard sense) to Template:Math for Template:Math. The Borel transform is

$ℬA(tz)\equiv \sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{k!}{t}^k=e^{zt},$

from which we obtain the Borel sum

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}ℬA(tz)dt=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{e}^{tz}dt=\frac{1}{1-z}$

which converges in the larger region Template:Math, giving an analytic continuation of the original series.

Considering instead the weak Borel transform, the partial sums are given by Template:Math, and so the weak Borel sum is

$\underset{t\to \mathrm{\infty }}{\lim }{e}^{-t}\sum _{n=0}^{\mathrm{\infty }}\frac{1-z^{n+1}}{1-z}\frac{t^n}{n!}=\underset{t\to \mathrm{\infty }}{\lim }\frac{e^{-t}}{1-z}(e^t-z{e}^{tz})=\frac{1}{1-z},$

where, again, convergence is on Template:Math. Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since for Template:Math,

$\underset{t\to \mathrm{\infty }}{\lim }{e}^{-t}(ℬA)(zt)=e^{t(z-1)}=0.$

An alternating factorial series

Consider the series

$A(z)=\sum _{k=0}^{\mathrm{\infty }}k!(-1{)}^k\cdot z^k,$

then Template:Math does not converge for any nonzero Template:Math. The Borel transform is

$ℬA(t)\equiv \sum _{k=0}^{\mathrm{\infty }}{(-t)}^k=\frac{1}{1+t}$

for Template:Math, which can be analytically continued to allTemplate:Math. So the Borel sum is

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}ℬA(tz)dt=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-t}}{1+tz}dt=\frac{1}{z}\cdot e^{1/z}\cdot \mathrm{\Gamma }(0,\frac{1}{z})$

(where Template:Math is the incomplete gamma function).

This integral converges for all Template:Math, so the original divergent series is Borel summable for all suchTemplate:Math. This function has an asymptotic expansion as Template:Math 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.

Again, since

$\underset{t\to \mathrm{\infty }}{\lim }{e}^{-t}(ℬA)(zt)=\underset{t\to \mathrm{\infty }}{\lim }\frac{e^{-t}}{1+zt}=0,$

for all Template:Math, the equivalence theorem ensures that weak Borel summation has the same domain of convergence, Template:Math.

An example in which equivalence fails

The following example extends on that given in Template:Harv. Consider

$A(z)=\sum _{k=0}^{\mathrm{\infty }}\left(\sum _{\ell =0}^{\mathrm{\infty }}\frac{(-1{)}^{\ell }(2\ell +2{)}^k}{(2\ell +1)!}\right)z^k.$

After changing the order of summation, the Borel transform is given by

$\begin{array}{cc}ℬA(t)& =\sum _{\ell =0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{((2\ell +2)t{)}^k}{k!}\right)\frac{(-1{)}^{\ell }}{(2\ell +1)!}\\ & =\sum _{\ell =0}^{\mathrm{\infty }}{e}^{(2\ell +2)t}\frac{(-1{)}^{\ell }}{(2\ell +1)!}\\ & =e^t\sum _{\ell =0}^{\mathrm{\infty }}(e^t{)}^{2\ell +1}\frac{(-1{)}^{\ell }}{(2\ell +1)!}\\ & =e^t\sin (e^t).\end{array}$

At Template:Math the Borel sum is given by

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^t\sin (e^{2t})dt=\underset{1}{\overset{\mathrm{\infty }}{\int }}\sin (u^2)du=\sqrt{\frac{\pi }{8}}-S(1)<\mathrm{\infty },$

where Template:Math is the Fresnel integral. Via the convergence theorem along chords, the Borel integral converges for all Template:Math (the integral diverges for Template:Math).

For the weak Borel sum we note that

$\underset{t\to \mathrm{\infty }}{\lim }{e}^{(z-1)t}\sin (e^{zt})=0$

holds only for Template:Math, and so the weak Borel sum converges on this smaller domain.

Existence results and the domain of convergence

Summability on chords

If a formal series Template:Math is Borel summable at Template:Math, then it is also Borel summable at all points on the chord Template:Math connecting Template:Math to the origin. Moreover, there exists a function Template:Math analytic throughout the disk with radius Template:Math such that

$\sum a_k{z}^k=a(z)(𝑩),$

for all Template:Math.

An immediate consequence is that the domain of convergence of the Borel sum is a star domain in Template:Math. More can be said about the domain of convergence of the Borel sum, than that it is a star domain, which is referred to as the Borel polygon, and is determined by the singularities of the series Template:Math.

The Borel polygon

Suppose that Template:Math has strictly positive radius of convergence, so that it is analytic in a non-trivial region containing the origin, and let Template:Math denote the set of singularities of Template:Math. This means that Template:Math if and only if Template:Math can be continued analytically along the open chord from 0 to Template:Math, but not to Template:Math itself. For Template:Math, let Template:Math denote the line passing through Template:Math which is perpendicular to the chord Template:Math. Define the sets

${\mathrm{\Pi }}_P=\{z\in ℂ:Oz\cap L_P=\varnothing \},$

the set of points which lie on the same side of Template:Math as the origin. The Borel polygon of Template:Math is the set

${\mathrm{\Pi }}_A=\mathrm{cl}\left(\bigcap _{P\in S_A}{\mathrm{\Pi }}_P\right).$

An alternative definition was used by Borel and Phragmén Template:Harv. Let $S\subset ℂ$ denote the largest star domain on which there is an analytic extension of Template:Math, then ${\mathrm{\Pi }}_A$ is the largest subset of $S$ such that for all $P\in {\mathrm{\Pi }}_A$ the interior of the circle with diameter OP is contained in $S$. Referring to the set ${\mathrm{\Pi }}_A$ as a polygon is something of a misnomer, since the set need not be polygonal at all; if, however, Template:Math has only finitely many singularities then ${\mathrm{\Pi }}_A$ will in fact be a polygon.

The following theorem, due to Borel and Phragmén provides convergence criteria for Borel summation.

Theorem Template:Harv.

The series Template:Math is Template:Math summable at all $z\in \mathrm{int}({\mathrm{\Pi }}_A)$, and is Template:Math divergent at all $z\in ℂ\setminus {\mathrm{\Pi }}_A$.

Note that Template:Math summability for $z\in \partial {\mathrm{\Pi }}_A$ depends on the nature of the point.

Example 1

Let Template:Math denote the Template:Math-th roots of unity, Template:Math, and consider

$\begin{array}{cc}A(z)& =\sum _{k=0}^{\mathrm{\infty }}({\omega }_1^k+\cdots +{\omega }_m^k)z^k\\ & =\sum _{i=1}^m\frac{1}{1-{\omega }_{i}z},\end{array}$

which converges on Template:Math. Seen as a function on Template:Math, Template:Math has singularities at Template:Math, and consequently the Borel polygon ${\mathrm{\Pi }}_A$ is given by the regular [[Regular polygon|Template:Math-gon]] centred at the origin, and such that Template:Math is a midpoint of an edge.

Example 2

The formal series

$A(z)=\sum _{k=0}^{\mathrm{\infty }}{z}^{2^k},$

converges for all $|z|<1$ (for instance, by the comparison test with the geometric series). It can however be shown that Template:Math does not converge for any point Template:Math such that Template:Math for some Template:Math. Since the set of such Template:Math is dense in the unit circle, there can be no analytic extension of Template:Math outside of Template:Math. Subsequently the largest star domain to which Template:Math can be analytically extended is Template:Math from which (via the second definition) one obtains ${\mathrm{\Pi }}_A=B(0,1)$. In particular one sees that the Borel polygon is not polygonal.

A Tauberian theorem

A Tauberian theorem provides conditions under which convergence of one summation method implies convergence under another method. The principal Tauberian theorem for Borel summation provides conditions under which the weak Borel method implies convergence of the series.

Theorem Template:Harv. If Template:Math is Template:Math summable at Template:Math, $\sum a_k{z}_0^k=a(z_0)(𝒘𝑩)$, and

$a_k{z}_0^k=O(k^{-1/2}),\mathrm{\forall }k\ge 0,$

then $\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}_0^k=a(z_0)$, and the series converges for all Template:Math.

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 Template:Harv. Some of the singularities of the Borel transform are related to instantons and renormalons in quantum field theory Template:Harv.

Generalizations

Borel summation requires that the coefficients do not grow too fast: more precisely, Template:Math has to be bounded by Template:Math for some Template:Math. There is a variation of Borel summation that replaces factorials Template:Math with Template:Math for some positive integer Template:Math, which allows the summation of some series with Template:Math bounded by Template:Math for some Template:Math. This generalization is given by Mittag-Leffler summation.

In the most general case, Borel summation is generalized by Nachbin resummation, which can be used when the bounding function is of some general type (psi-type), instead of being exponential type.

See also

• Abel summation
• Abel's theorem
• Abel–Plana formula
• Euler summation
• Cesàro summation
• Lambert summation
• Laplace transform
• Nachbin resummation
• Abelian and tauberian theorems
• Van Wijngaarden transformation

References

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