Borel-Summierung

In mathematics, Borel summation is a summation method for divergent series, introduced by Vorlage:Harvs. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.

Definition

Throughout let A(z) denote a formal power series

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

and define the Borel transform of A to be its equivalent exponential series

${\mathcal {B}}A(t)\equiv \sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}t^{k}.$

Borel summation

Suppose that the Borel transform converges to an analytic function of t 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), the Borel sum of A is given by

$\int _{0}^{\infty }e^{-t}{\mathcal {B}}A(tz)\,dt.$

If the integral converges at z ∈ C to some a(z), we say that the Borel sum of A converges at z, and write ${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})$.

Weak Borel summation

Let A_n(z) denote the partial sum

$A_{n}(z)=\sum _{k=0}^{n}a_{k}z^{k}.$

A weaker form of Borel's summation method defines the Borel sum of A to be

$\lim _{t\rightarrow \infty }e^{-t}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}A_{n}(z).$

If this converges at z ∈ C to some a(z), we say that the weak Borel sum of A converges at z, and write ${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})$.

Basic properties

Regularity

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

$\sum _{k=0}^{\infty }a_{k}z^{k}=A(z)<\infty \quad \Rightarrow \quad {\textstyle \sum }a_{k}z^{k}=A(z)\,\,({\boldsymbol {B}},\,{\boldsymbol {wB}}).$

Regularity of (B) is easily seen by a change in order of integration: if A(z) is convergent at z,then

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

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).

Equivalence of Borel and weak Borel summation

Any series A(z) that is weak Borel summable at z ∈ C is also Borel summable at z. 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 (Vorlage:Harv).

Let A(z) be a formal power series, and fix z ∈ C, then:

1. If ${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})$, then ${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})$.
2. If ${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})$, and $\lim _{t\rightarrow \infty }e^{-t}{\mathcal {B}}A(zt)=0,$ then ${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})$.

Relationship to other summation methods

• (B) is the special case of Mittag-Leffler_summation with α = 1.
• (wB) can be seen as the limiting case of generalized Euler summation method (E,q) in the sense that as q → ∞ the domain of convergence of the (E,q) method converges up to the domain of convergence for (B).

Examples

The geometric series

Consider the geometric series

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

which converges (in the standard sense) to 1/(1 − z) for |z| < 1. The Borel transform is

${\mathcal {B}}A(t)\equiv \sum _{k=0}^{\infty }{\frac {1}{k!}}t^{k}=e^{t},$

from which we obtain the Borel sum

$\int _{0}^{\infty }e^{-t}{\mathcal {B}}A(tz)\,dt=\int _{0}^{\infty }e^{-t}e^{tz}\,dt={\frac {1}{1-z}}$

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

Considering instead the weak Borel transform, the partial sums are given by A_N(z) = (1-z^N+1)/(1-z), and so the weak Borel sum is

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

where, again, convergence is on Re(z) < 1. Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since for Re(z) < 1

$\lim _{t\rightarrow \infty }e^{-t}({\mathcal {B}}A)(zt)=e^{t(z-1)}=0.$

An alternating factorial series

Consider the series

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

then A(z) does not converge for any nonzero z ∈  C. The Borel transform is

${\mathcal {B}}A(t)\equiv \sum _{k=0}^{\infty }\left(-1\cdot t\right)^{k}={\frac {1}{1+t}}$

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

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

(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.

Again, since

$\lim _{t\rightarrow \infty }e^{-t}({\mathcal {B}}A)(zt)=\lim _{t\rightarrow \infty }{\frac {e^{-t}}{1+zt}}=0,$

for all z, the equivalence theorem ensures that weak Borel summation has the same domain of convergence, z ≥ 0.

An example in which equivalence fails

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

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

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

${\begin{aligned}{\mathcal {B}}A(t)&=\sum _{l=0}^{\infty }\left(\sum _{k=0}^{\infty }{\frac {{\big (}(2l+2)t{\big )}^{k}}{k!}}\right){\frac {(-1)^{l}}{(2l+1)!}}\\&=\sum _{l=0}^{\infty }e^{(2l+2)t}{\frac {(-1)^{l}}{(2l+1)!}}\\&=e^{t}\sum _{l=0}^{\infty }{\big (}e^{t}{\big )}^{2l+1}{\frac {(-1)^{l}}{(2l+1)!}}\\&=e^{t}\sin \left(e^{t}\right).\end{aligned}}$

At z = 2 the Borel sum is given by

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

where S(x) is the Fresnel integral. Via the convergence theorem along chords, the Borel integral converges for all z ≤ 2 (clearly the integral diverges for z > 2).

For the weak Borel sum we note that

$\lim _{t\rightarrow \infty }e^{(z-1)t}\sin \left(e^{zt}\right)=0$

holds only for z < 1, 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 A(z) is Borel summable at z₀ ∈ C, then it is also Borel summable at all points on the chord Oz₀ connecting z₀ to the origin. Moreover, there exists a function a(z) analytic throughout B(O,z₀) such that

${\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}}),$

for all z = θz₀, θ ∈ [0,1].

An immediate consequence is that the domain of convergence of the Borel sum is a star domain in C. 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 A(z).

The Borel polygon

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 S_A denote the set of singularities of A. For P ∈ S_A, let L_P denote the line passing through P which is perpendicular to the chord OP. Define the sets

$\Pi _{P}=\{z\in \mathbb {C} \,\colon \,Oz\cap L_{P}=\varnothing \},$

the set of points which lie on the same side of L_P as the origin. The Borel polygon of A is the set

$\Pi _{A}={\text{cl}}{\Big (}\bigcap _{P\in S_{A}}\Pi _{P}{\Big )}.$

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

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

Theorem Vorlage:Harv.

The series A(z) is (B) summable at all $z\in {\text{int}}(\Pi _{A})$, and is (B) divergent at all $z\in \mathbb {C} \backslash \Pi _{A}$.

Note that (B) summability for $z\in \partial \Pi _{A}$ depends on the nature of the point.

Example 1

Let ω_i ∈ C denote the m-th roots of unity, i =1,...m, and consider

${\begin{aligned}A(z)&=\sum _{k=0}^{\infty }(\omega _{1}^{k}+\ldots +\omega _{m}^{k})z^{k}\\&=\sum _{i=1}^{m}{\frac {1}{1-\omega _{i}z}},\end{aligned}}$

which converges on B(0,1) ⊂ C. Seen as a function on C, A(z) has singularities at S_A = {ω_i : i = 1,...m}, and consequently the Borel polygon $\Pi _{A}$ is given by the regular m-gon centred at the origin, and such that 1 ∈ C is a midpoint of an edge.

Example 2

The formal series

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

converges for all $|z|<1$ (for instance, by the comparison test with the geometric series). It can however be shown that A does not converge for all points z ∈ C with z²ⁿ = 1. 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 $\Pi _{A}=B(0,1)$. In particular we see that the Borel polygon is in fact 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 Vorlage:Harv. :If A is (wB) summable at z₀ ∈ C, ${\textstyle \sum }a_{k}z_{0}^{k}=a(z_{0})\,({\boldsymbol {wB}})$, and

$a_{k}z_{0}^{k}=O\left(k^{-{\textstyle {\frac {1}{2}}}}\right),\qquad \forall k\geq 0,$

then $\sum _{k=0}^{\infty }a_{k}z_{0}^{k}=a(z_{0})$, and the series converges for all |z|<|z₀|.

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

See also

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

References

• Vorlage:Citation
• Vorlage:Citation
• Vorlage:Citation
• Vorlage:Citation
• Vorlage:Citation
• Vorlage:Citation
• Vorlage:Eom