Borel integral with a simple pole
= Borel integral with a simple pole
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>.