Integral Moments of the Lambert W Function Near the Branch Point: Exact Values and Failure of the Uniform Airy Approximation for Global Integrals

We compute exact numerical values of the integral moments I1(n), I2(n), and I3(n) of the Lambert W function for n = 0,...,5, where the integrals probe the behaviour of W0(z) raised to the n-th power over domains that include or are adjacent to the branch point z = -1/e. We then test the uniform Airy approximation for W(z) near z = -1/e by substituting it into the same integrands and comparing with the exact values. The approximation reproduces the n=0 case exactly but fails dramatically for n >= 1, with relative errors ranging from 285% to more than 26,000%. This negative result demonstrates that the uniform Airy approximation, designed for the local behaviour of the function near the branch point, cannot be extrapolated to global integral quantities. The exact numerical tables serve as benchmarks for future analytical or numerical work.