INFINITELY MANY STOKES SMOOTHINGS IN THE GAMMA-FUNCTION

The Stokes lines for GAMMA-(z) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for ln GAMMA-(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 GAMMA-(z). Corresponding to each small exponential is a separate component asymptotic series int he expansion for ln GAMMA-(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 substractions from ln GAMMA-(z), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.