Borel summation

In mathematics, a Borel summation is a generalisation of the usual notion of summation of a series. In particular it gives a definition of a quantity that in many ways behaves formally like the sum, even if the series is in fact divergent.

Definition

Let

:$y\; =\; sum\_\{k\; =\; 0\}^infty\; y\_kz^\{-k\}$

be a formal power series in "z".

Define the "Borel transform" $mathcal\{B\}y$ of $y$ by:$sum\_\{k=0\}^infty\; frac\{y\_k\}\{(k-1)!\}t^\{k-1\}.$

Suppose that

# $scriptstylemathcal\{B\}y$ has a nonzero radius of convergence as a function of "t";
# $scriptstylemathcal\{B\}y$ can be analytically continued to a function $scriptstylewidehat\{y\}(t)$ on all of the positive real line;
# $scriptstylewidehat\{y\}(t)$ grows at most exponentially along the positive real line.

Then the Borel sum of "y" is given by the Laplace transform of $scriptstylewidehat\{y\}(t)$. This function is guaranteed to exist by condition (3) above.

Discussion

The Borel sum of a series is the Laplace transform of the sum of the term-by-term inverse Laplace transform of the original series. If the Laplace transform of an infinite series were equal to the sum of its term-by-term Laplace transform then the Borel sum would be equal to the usual sum. The Borel sum is defined in many situations where the sum isn't defined. Speaking nonrigorously, it allows us to attach a meaning to the 'sum' of certain types of divergent series. Borel summation is an example of a moment constant method for summing series.

Applications

Borel summation finds application in perturbation theory where physicists frequently require the sum of a series even though it is divergent.

A direct extension of Borel resummation from series (discrete) to integrals (continuous) can be given in the form

: $int\_\{0\}^\{infty\}\; s^\{-x\}f(x),dx\; ightarrow\; sint\_\{0\}^\{infty\}\; int\_\{0\}^\{infty\}\; frac\{f(x)t^\{x\{Gamma\; (x+1)\; \}exp(-st),dt,dx\; =\; frac\{F(ln(s))\}\{ln(s)\}$

where "F"("s") is the Laplace transform of "f"("x"). This is used to give a finite meaning to Fourier integrals of the type

: $int\_\{-infty\}^\{infty\}\; f(x)e^\{iomega\; x\},dx.$

History

Nicholas M. Katz records an anecdote from Émile Borel's youth:

References