Approximaths

Borel–Écalle summation

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Consider the archetypal divergent series

\displaystyle \sum_{n=0}^\infty n!\, x^n.

In a previous post we have seen the Borel summation of a_1 + a_2 + ... + a_n+ ...

\displaystyle \begin{aligned} B &:= \sum_{n=0}^{\infty} \frac{\int_{0}^{\infty} e^{-t}t^{n}dt}{n!} a_n \\ &= \int_{0}^{\infty}e^{-t} \sum_{n=0}^{\infty} \frac{t^n}{n!}a_ndt \end{aligned}

Step 1 – Ordinary Borel summation

\displaystyle \sum_{n=0}^{\infty} \frac{t^n}{n!} a_n = \sum_{n=0}^{\infty} \frac{t^n}{n!} n!x^n = \sum_{n=0}^\infty (x t)^n = \frac{1}{1 - x t}

The ordinary Borel summation fails for x > 0 the analytic continuation of \mathcal{B}(t) has a simple pole at t = 1/x lying on the positive real axis. The ordinary Borel integral

\displaystyle \int_0^\infty e^{-t} \frac{1}{1 - x t}\, dt

therefore diverges for all x > 0 (the pole blocks the integration path).

Step 2 – Borel–Écalle summation
Define the two lateral Borel transforms by deforming the contour slightly above (+) or below (-) the real axis:

\displaystyle B^\pm(x) = \int_0^{\infty e^{\pm i 0}} e^{-t} \frac{1}{1 - x t}\, dt.

The notation ∞e±i0 means that the upper limit of integration is taken to infinity along a ray that approaches the positive real axis from above (angle +0) or from below (angle −0). This slight contour deformation is necessary when the integrand has a singularity (here at t = 1/x) on the positive real axis itself, which would cause the ordinary Borel integral to be ill-defined.

These integrals exist, and

\displaystyle B^+(x) - B^-(x) = \frac{2\pi i \, e^{-1/x}}{x}.

The Borel–Écalle summation is defined by

\displaystyle y(x) := \frac{B^+(x) + B^-(x)}{2}

And in this case:

\displaystyle y(x) = -e^{1/x} E_i\!\left(\frac{1}{x}\right)

where E_i(z) = \int_z^\infty \frac{e^{-u}}{u}\, du is the exponential integral function. Thus, despite the divergence of the original series and the failure of ordinary Borel summation due to the pole on the integration path, the Borel–Écalle median summation recovers the exact analytic continuation on the positive real axis.

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