Examples
The series
converges to 1/(1 − z) for |z| < 1. The Borel transform is
So the Borel sum is
which converges to 1/(1 − z) in the larger region Re(z) < 1, giving an analytic continuation of the original series.
The series
does not converge for any nonzero z.
The Borel transform is
for |t| < 1, and this can be analytically continued to all t ≥ 0. So the Borel sum is
(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.
Read more about this topic: Borel Summation
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)