Mellin-Barnes regularization, Borel summation and the Bender-Wu asymptotics for the anharmonic oscillator

We introduce the numerical technique of Mellin-Barnes integral regularization, which can be used to evaluate both convergent and divergent series. The technique is shown to be numerically equivalent to the corresponding results obtained by Borel summation. Both techniques are then applied to the Bender-Wu formula, which represents an asymptotic expansion for the energy levels of the anharmonic oscillator We find that this formula is unable to give accurate values for the ground-state energy, particularly when the coupling is greater than 0.1. As a consequence, the inability of the Bender-Wu formula to yield exact values for the energy level of the anharmonic oscillator cannot be attributed to its asymptotic nature.