Some remarks on the theorems of Wright and Braaksma on the Wright function ${}_pΨ_q(z)$

Abstract: We carry out a numerical investigation of the asymptotic expansion of the so-called Wright function ${}_p\Psi_q(z)$ (a generalised hypergeometric function) in the case when exponentially small terms are present. This situation is covered by two theorems of Wright and Braaksma. We demonstrate that a more precise understanding of the behaviour of ${}_p\Psi_q(z)$ is obtained by taking into account the Stokes phenomenon.