Verifying the inverse Laplace transform of total expected stock-outs when demand is Poisson

This note aims to verify a Laplace transform pair, previously published without proof, concerning the expected stock-out that may occur in a production-inventory systems when demand is Poisson, and the time horizon is finite. Stock-out, or "backlog", is understood as the non-negative difference between cumulative demand, a stochastic variable, and production over a finite time interval. The hoped for outcome is greater clarity on how the Laplace transform pair of interest might be arrived at, and a possible improvement in terms of accuracy on the original result.