We obtain the rigorous uniform asymptotics of a particular integral where a stationary point is close to an endpoint. There exists a general method introduced by Bleistein for obtaining uniform asymptotics in this situation. However, this method does not provide rigorous estimates for the error. Indeed, the method of Bleistein starts with a change of variables, which implies that the parameter governing how close the stationary point is to the endpoint appears in several parts of the integrand, and this means that one cannot obtain general error bounds. By adapting the above method to our particular integral, we obtain rigorous uniform leading-order asymptotics. We also give a rigorous derivation of the asymptotics to all orders of the same integral; the novelty of this second approach is that it does not involve a global change of variables.
Fernandez, A., Spence, E., & Fokas, A. S. (2018). Uniform asymptotics as a stationary point approaches an endpoint. IMA Journal of Applied Mathematics, 83(1), 204-242. https://doi.org/10.1093/imamat/hxx042