Infinitely Many Stokes Smoothings in the Gamma Function

The Stokes lines for Г(z) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for In Г(z) have the same phase. On crossing these lines from the right to the left half-plane, infinitely many subdominant exponentials appear, rather than the usual one. The exponentials increase in magnitude towards the negative real axis (anti-Stokes line), where they add to produce the poles of Г(z). Corresponding to each small exponential is a separate component asymptotic series in the expansion for In Г(z). If each is truncated near its least term, its exponential switches on smoothly across the Stokes lines according to the universal error-function law. By appropriate subtractions from In Г(z), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.