In mathematics, Borel summation is a summation method for divergent series, introduced by . 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.
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 denote a formal power series
A(z)=
infty | |
\sum | |
k=0 |
k, | |
a | |
kz |
and define the Borel transform of to be its equivalent exponential series
l{B}A(t)\equiv
infty | |
\sum | |
k=0 |
ak | |
k! |
tk.
Let denote the partial sum
An(z)=
n | |
\sum | |
k=0 |
akzk.
A weak form of Borel's summation method defines the Borel sum of to be
\limt → inftye-t
infty | |
\sum | |
n=0 |
tn | |
n! |
An(z).
If this converges at to some function, we say that the weak Borel sum of converges at, and write
{style\sum}
k | |
a | |
kz |
=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 is given by
infty | |
\int | |
0 |
e-tl{B}A(tz)dt.
If the integral converges at to some, we say that the Borel sum of converges at, and write
{style\sum}
k | |
a | |
kz |
=a(z)(\boldsymbolB)
This is similar to Borel's integral summation method, except that the Borel transform need not converge for all, but converges to an analytic function of near 0 that can be analytically continued along the positive real axis.
The methods and are both regular summation methods, meaning that whenever converges (in the standard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.
infty | |
\sum | |
k=0 |
akzk=A(z)<infty ⇒ {style\sum}
k | |
a | |
kz |
=A(z)(\boldsymbol{B},\boldsymbol{wB}).
Regularity of is easily seen by a change in order of integration, which is valid due to absolute convergence: if is convergent at, then
A(z)=
infty | |
\sum | |
k=0 |
akzk=
infty | |
\sum | |
k=0 |
ak\left(
infty | |
\int | |
0 |
e-ttkdt\right)
zk | |
k! |
=
infty | |
\int | |
0 |
e-t
infty | |
\sum | |
k=0 |
ak
(tz)k | |
k! |
dt,
where the rightmost expression is exactly the Borel sum at .
Regularity of and imply that these methods provide analytic extensions to .
Any series that is weak Borel summable at is also Borel summable at . 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 .
Let be a formal power series, and fix, then:
{style\sum}
k | |
a | |
kz |
=a(z)(\boldsymbol{wB})
k | |
{style\sum}a | |
kz |
=a(z)(\boldsymbol{B})
{style\sum}
k | |
a | |
kz |
=a(z)(\boldsymbol{B})
\limte-tlBA(zt)=0,
{style\sum}
k | |
a | |
kz |
=a(z)(\boldsymbol{wB})
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 gives conditions for a function to be the Borel sum of its asymptotic series. Suppose that is a function satisfying the following conditions:
|f(z)-a0-a1z- … -an-1zn-1|
Cn+1n!|z|n
Then Watson's theorem says that in this region 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 for 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 is analytic in the interior of the sector, and in this region for all, then is zero provided that the series 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 for some constant . More generally one can define summation methods slightly stronger than Borel's by taking the numbers to be slightly larger, for example or . 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 has the asymptotic series with an error bound of the form above in the region for any, but is not given by the Borel sum of its asymptotic series. This shows that the number in Watson's theorem cannot be replaced by any smaller number (unless the bound on the error is made smaller).
Consider the geometric series
A(z)=
infty | |
\sum | |
k=0 |
zk,
which converges (in the standard sense) to for . The Borel transform is
l{B}A(tz)\equiv
infty | |
\sum | |
k=0 |
zk | |
k! |
tk=ezt,
from which we obtain the Borel sum
infty | |
\int | |
0 |
e-tl{B}A(tz)dt=
infty | |
\int | |
0 |
e-tetzdt=
1 | |
1-z |
which converges in the larger region, giving an analytic continuation of the original series.
Considering instead the weak Borel transform, the partial sums are given by, and so the weak Borel sum is
\limte-t
infty | |
\sum | |
n=0 |
1-zn+1 | |
1-z |
tn | |
n! |
=\limt
e-t | |
1-z |
(et-zetz)=
1 | |
1-z |
,
where, again, convergence is on . Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since for,
\limte-t(l{B}A)(zt)=et(z-1)=0.
Consider the series
A(z)=
infty | |
\sum | |
k=0 |
k!(-1)k ⋅ zk,
then does not converge for any nonzero . The Borel transform is
l{B}A(t)\equiv
infty | |
\sum | |
k=0 |
\left(-t\right)k=
1 | |
1+t |
for, which can be analytically continued to all. So the Borel sum is
infty | |
\int | |
0 |
e-tl{B}A(tz)dt=
infty | |
\int | |
0 |
e-t | |
1+tz |
dt=
1 | |
z |
⋅ e1/z ⋅ \Gamma\left(0,
1 | |
z |
\right)
(where is the incomplete gamma function).
This integral converges for all, so the original divergent series is Borel summable for all such. This function has an asymptotic expansion as 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
\limte-t(lBA)(zt)=\limt
e-t | |
1+zt |
=0,
for all, the equivalence theorem ensures that weak Borel summation has the same domain of convergence, .
The following example extends on that given in . Consider
A(z)=
infty | |
\sum | |
k=0 |
\left(
infty | |
\sum | |
\ell=0 |
(-1)\ell(2\ell+2)k | |
(2\ell+1)! |
\right)zk.
After changing the order of summation, the Borel transform is given by
\begin{align} lBA(t)&=
infty | |
\sum | |
\ell=0 |
\left(
infty | |
\sum | |
k=0 |
((2\ell+2)t)k | |
k! |
\right)
(-1)\ell | |
(2\ell+1)! |
\\ &=
infty | |
\sum | |
\ell=0 |
e(2\ell+2)t
(-1)\ell | |
(2\ell+1)! |
\\ &=et
infty | |
\sum | |
\ell=0 |
(et)2\ell+1
(-1)\ell | |
(2\ell+1)! |
\\ &=et\sin(et). \end{align}
At the Borel sum is given by
infty | |
\int | |
0 |
et\sin(e2t)dt=
infty | |
\int | |
1 |
\sin(u2)du=\sqrt{
\pi | |
8 |
where is the Fresnel integral. Via the convergence theorem along chords, the Borel integral converges for all (the integral diverges for).
For the weak Borel sum we note that
\limte(z-1)t\sin(ezt)=0
holds only for, and so the weak Borel sum converges on this smaller domain.
If a formal series is Borel summable at, then it is also Borel summable at all points on the chord connecting to the origin. Moreover, there exists a function analytic throughout the disk with radius such that
{style\sum}
k | |
a | |
kz |
=a(z)(\boldsymbolB),
for all .
An immediate consequence is that the domain of convergence of the Borel sum is a star domain in . 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 .
Suppose that has strictly positive radius of convergence, so that it is analytic in a non-trivial region containing the origin, and let denote the set of singularities of . This means that if and only if can be continued analytically along the open chord from 0 to, but not to itself. For, let denote the line passing through which is perpendicular to the chord . Define the sets
\PiP=\{z\inC\colonOz\capLP=\varnothing\},
the set of points which lie on the same side of as the origin. The Borel polygon of is the set
\PiA=\operatorname{cl}\left(
cap | |
P\inSA |
\PiP\right).
An alternative definition was used by Borel and Phragmén . Let
S\subsetC
\PiA
S
P\in\PiA
S
\PiA
\PiA
The following theorem, due to Borel and Phragmén provides convergence criteria for Borel summation.
Theorem .
The series is summable at all
z\in\operatorname{int}(\PiA)
z\inC\setminus\PiA
Note that summability for
z\in\partial\PiA
Let denote the -th roots of unity,, and consider
\begin{align} A(z)&=
infty | |
\sum | |
k=0 |
k | |
(\omega | |
1 |
+ … +
k)z | |
\omega | |
m |
k\\ &=
m | |
\sum | |
i=1 |
1 | |
1-\omegaiz |
, \end{align}
which converges on . Seen as a function on, has singularities at, and consequently the Borel polygon
\PiA
The formal series
A(z)=
infty | |
\sum | |
k=0 |
2k | |
z |
,
converges for all
|z|<1
\PiA=B(0,1)
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 . If is summable at,
{style\sum}akz
k | |
0 |
=a(z0)(\boldsymbol{wB})
akz
k | |
0 |
=O(k-1/2), \forallk\geq0,
then
infty | |
\sum | |
k=0 |
akz
k | |
0 |
=a(z0)
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 . Some of the singularities of the Borel transform are related to instantons and renormalons in quantum field theory .
Borel summation requires that the coefficients do not grow too fast: more precisely, has to be bounded by for some . There is a variation of Borel summation that replaces factorials with for some positive integer, which allows the summation of some series with bounded by for some . 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.