Borel summation (${\int }_0^{\infty }{e}^{-zt}\mathcal{B}(t)dt$)

For a formal inverse-power series ${\sum }_{n\ge 0}{a}_n{z}^{-n-1}$, divide its coefficients by $n!$ to form a Borel transform $\sum a_n{t}^n/n!$ and integrate it against $e^{-zt}$. Singularities on the integration ray require a specified contour prescription. Different lateral contours can produce exponentially small differences and the Stokes phenomenon.

If a Borel summation contour meets a simple pole, an upper or lower indentation fixes the exponentially small contribution. For the transform $(1-t^2{)}^{-1}$ on the positive ray, the relevant pole is $t=1$. A lower indentation gives the Cauchy principal value minus $i\pi e^{-z}/2$. Finite geometric expansion gives an exact remainder integral, which enables optimal truncation and error-function smoothing of a Stokes multiplier.