Borel summation

Let A_n(z) denote the partial sum

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

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

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

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

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

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

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

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

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.

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

Regularity of (B) is easily seen by a change in order of integration, which is valid due to absolute convergence: if A(z) is convergent at z, then

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

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).

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 ((Hardy 1992, 8.5)).

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

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

• (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).[1]

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:

• f is holomorphic in some region |z| < R, |arg(z)| < π/2 + ε for some positive R and ε.
• In this region f has an asymptotic series a₀ + a₁z + ... with the property that the error

${\displaystyle |f(z)-a_{0}-a_{1}z-\cdots -a_{n-1}z^{n-1}|}$

is bounded by

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

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.

Slightly more generally, f is still determined by its asymptotic series if the n! in the error estimate above is replaced by kn! provided the condition |arg(z)| < π/2 + ε is replaced by |arg(z)| < kπ/2 + ε. This is in some sense best possible, as there are counterexamples if the number kπ/2 is replaced by any smaller number.

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)| < |b_nz|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 b_n =cn for some constant c. More generally one can define summation methods slightly stronger than Borel's by taking the numbers b_n to be slightly larger, for example b_n = cnlog n or b_n =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.

The function f(z) = exp(–1/z) has the asymptotic series 0 + 0z+ ... with an error bound of the form above in the region |arg(z)| < θ for any θ < π/2, but is not given by the Borel sum of its asymptotic series. This shows that the number π/2 in Watson's theorem cannot be replaced by any smaller number (unless the bound on the error is made smaller).

Consider the geometric series

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

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

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

from which we obtain the Borel sum

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

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

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

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,

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

Consider the series

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

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

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

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

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

(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

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

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

The following example extends on that given in (Hardy 1992, 8.5). Consider

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

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

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

At z = 2 the Borel sum is given by

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

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

For the weak Borel sum we note that

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

holds only for z < 1, and so the weak Borel sum converges on this smaller domain.

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 the disk with radius Oz₀ such that

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

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).

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. This means that P ∈ S_A if and only if A can be continued analytically along the open chord from 0 to P, but not to P itself. For P ∈ S_A, let L_P denote the line passing through P which is perpendicular to the chord OP. Define the sets

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

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

${\displaystyle \Pi _{A}=\operatorname {cl} \left(\bigcap _{P\in S_{A}}\Pi _{P}\right).}$

An alternative definition was used by Borel and Phragmén (Sansone & Gerretsen 1960, 8.3). Let ${\displaystyle S\subset \mathbb {C} }$ denote the largest star domain on which there is an analytic extension of A, then ${\displaystyle \Pi _{A}}$ is the largest subset of ${\displaystyle S}$ such that for all ${\displaystyle P\in \Pi _{A}}$ the interior of the circle with diameter OP is contained in ${\displaystyle S}$. Referring to the set ${\displaystyle \Pi _{A}}$ 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 ${\displaystyle \Pi _{A}}$ will in fact be a polygon.

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

Theorem (Hardy 1992, 8.8).

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

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

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

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

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 ${\displaystyle \Pi _{A}}$ is given by the regular m-gon centred at the origin, and such that 1 ∈ C is a midpoint of an edge.

The formal series

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

converges for all ${\displaystyle |z|<1}$ (for instance, by the comparison test with the geometric series). 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 ${\displaystyle \Pi _{A}=B(0,1)}$. In particular one sees that the Borel polygon is not polygonal.

A Tauberian theorem provides conditions under which convergence of one summation method implies convergence under another method. The principal Tauberian theorem[1] for Borel summation provides conditions under which the weak Borel method implies convergence of the series.

Theorem (Hardy 1992, 9.13). If A is (wB) summable at z₀ ∈ C, ${\displaystyle {\textstyle \sum }a_{k}z_{0}^{k}=a(z_{0})\,({\boldsymbol {wB}})}$, and

${\displaystyle a_{k}z_{0}^{k}=O(k^{-1/2}),\qquad \forall k\geq 0,}$

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