= Borel transform of a factorially divergent series
{c}
{title2=$\mathcal B(t)=\sum_{r\ge1}a_rt^{r-1}/\Gamma(r)$}
= Borel transform
{c}
{synonym}
Removing factorial growth from the coefficients of a formal inverse-power series produces its Borel transform. When continuation and exponential growth bounds allow it, $\int_{\mathcal C}e^{-zt}\mathcal B(t)\,dt$ reconstructs the series by <Laplace transform>. A singularity on the integration direction requires a lateral contour prescription. A <simple pole> contributes a <residue> when that prescription is changed, explaining how <Stokes phenomenon> arises from otherwise identical algebraic expansions.
Back to article page
